aa r X i v : . [ m a t h . R A ] F e b FINITE Σ -RICKART MODULES GANGYONG LEE AND MAURICIO MEDINA-B ´ARCENAS
Abstract.
In this article, we study the notion of a finite Σ-Rickart module, as a moduletheoretic analogue of a right semi-hereditary ring. A module M is called finite Σ -Rickart if every finite direct sum of copies of M is a Rickart module. It is shown that any directsummand and any direct sum of copies of a finite Σ-Rickart module are finite Σ-Rickartmodules. We also provide generalizations in a module theoretic setting of the most commonresults of semi-hereditary rings. Also, we have a characterization of a finite Σ-Rickartmodule in terms of its endomorphism ring. In addition, we introduce M -coherent modulesand provide a characterization of finite Σ-Rickart modules in terms of M -coherent modules.At the end, we study when Σ-Rickart modules and finite Σ-Rickart modules coincide.Examples which delineate the concepts and results are provided. Key Words: semi-hereditary ring, finite Σ-Rickart module, Rickart module, M -coherentmodule, f-injective, M -pure epimorphism1. Introduction
After hereditary rings were introduced by Kaplansky in the earliest 50’s, many math-ematicians studied semi-hereditary rings as a natural generalization of hereditary rings.Recall that a ring R is said to be right semi-hereditary if every finitely generated rightideal of R is projective. In [20] L. Small gives an example of a ring R which is right semi-hereditary but R is not right hereditary. Following the research on hereditary rings, manycharacterizations of semi-hereditary rings also have been made. For instance, in [2] is provedthat a ring R is right semi-hereditary if and only if every finitely generated submodule of aprojective R -module is projective. Later, in [15, Theorem 2] is shown that a ring R is rightsemi-hereditary if and only if factor modules of absolutely pure R -modules are absolutelypure (also, if and only if factor modules of injective R -modules are f-injective). Chase in [3]characterizes a right semi-hereditary ring as a right coherent ring whose right ideals are flat.Very close to right semi-hereditary rings are right Rickart rings. A ring R is said to be rightRickart if the right annihilator of any element of R is generated by an idempotent. Smallin [21, Proposition] proves that a ring R is right semi-hereditary if and only if Mat n ( R )is a right Rickart ring for every positive integer n . In 2012 Lee, Rizvi, and Roman in [13]extend Small’s result with the theory of Rickart modules. A right R -module M is called Rickart if Ker ϕ is a direct summand of M for any ϕ ∈ End R ( M ). They prove that a ring R is right semi-hereditary if and only if R ( n ) is a Rickart module for every positive integer n [13, Theorem 3.6]. Inspired by the last result, in this paper we define finite Σ-Rickartmodules. A right R -module M is called finite Σ -Rickart if M ( n ) is a Rickart module forevery n >
0. We present many properties of these modules extending those for right semi-hereditary rings. For a module M , we will focus on finitely M -generated modules as a moregeneral concept of finitely generated modules. We study a finite Σ-Rickart module M usingthe class , add( M ), which is the analogue to that of finitely generated projective modules, Mathematics Subject Classification.
Primary 16D40; 16D50; 16E50; 16E60; 16S50.THIS IS A MODIFIED AND EXTENDED VERSION OF THE VERSION SUBMITTED FORPUBLICATION. add( R ), in the case of the ring R . To get the module theoretic version of Megibben’s result[15, Theorem 2] mentioned above, we introduce the class F M : in the case of the ring M = R , F R is the class of absolutely pure modules. We will compare F M with the class E M whichis introduced in [11]. Note that E R is the class of injective modules.After the introduction and some preliminary background, in Section 2, finite Σ-Rickartmodules are defined, some examples are presented, and the general properties of thesemodules are studied. It is proved that direct summands and finite direct sums of copies ofa finite Σ-Rickart module inherit the properties (Lemma 2.4). It is shown that M is finiteΣ-Rickart if and only if every finitely M -generated submodule of an element in add( M )has D condition (Theorem 2.13). We introduce the class F M for a right R -module M and characterize finite Σ-Rickart modules in terms of this new class (Theorem 2.26) whichis a module theoretic version of [15, Theorem 2]. At the end of the section we provide acharacterization of an endoregular module in terms of a finite Σ-Rickart module as well as acharacterization of von Neumann regular rings as a corollary (Theorem 2.29 and Corollary2.30, respectively).Our focus in Section 3 is on the endomorphism ring of a finite Σ-Rickart module. Weintroduce the concept of M -coherent modules and we link it under intrinsically projectivemodules. That is, when M is intrinsically projective, two characterizations for an M -coherent module M in terms of the intersection property of finitely M -generated submodulesof M (Theorem 3.10) and in terms of when End R ( M ) is a right coherent ring (Theorem 3.15)are provided. These results will help us to characterize a finite Σ-Rickart module in termsof its endomorphism ring. A module M is finite Σ-Rickart if and only if S = End R ( M )is a right semi-hereditary ring and S M is flat if and only if M is intrinsically projective, M -coherent and all right S -ideals are flat (Theorem 3.20).When a module S M is flat where S = End R ( M ) in Section 4, we prove that M ∈ E M ifand only if S is a right self-injective ring, and M ∈ F M if and only if S is a right f-injectivering (Corollary 4.3). Also, for M = L ni =1 H ( ℓ i ) i , if H i is an indecomposable endoregularmodule and H i is H j -Rickart for all 1 ≤ i, j ≤ n then End R ( M ) is a semiprimary PWD(Corollary 4.8). Therefore as an application, if M is a finite Σ-Rickart module and P is anysimple module such that Hom R ( M, P ) = 0 then M ( ℓ ) ⊕ P ( n ) is a finite Σ-Rickart module forany ℓ, n > R is an associative ring with unity and M is a unitary right R -module. For a right R -module M , S = End R ( M ) will denote the endomorphism ringof M ; thus M can be viewed as a left S - right R -bimodule. For ϕ ∈ S , Ker ϕ and Im ϕ stand for the kernel and the image of ϕ , respectively. The notations N ≤ M , N E M , N ≤ ess M , and N ≤ ⊕ M mean that N is a submodule, a fully invariant submodule, anessential submodule, and a direct summand of M , respectively. We use M ( n ) to denotethe direct sum of n copies of M . By Q , Z , and N we denote the set of rational, integer,and natural numbers, respectively. For 1 < n ∈ N , Z n denotes the Z -module Z /n Z . Wealso denote r R ( N ) = { r ∈ R | N r = 0 } and l S ( N ) = { ϕ ∈ S | ϕN = 0 } for N ≤ M , and r S ( I ) = { ϕ ∈ S | ϕI = 0 } for I S ≤ S .In [12] was introduced the concept of Rickart modules and were presented many propertiesof them. Definition 1.1.
A right R -module M is called Rickart if Ker ϕ is a direct summand of M for every endomorphism ϕ ∈ End R ( M ) [12]. M is called endoregular if End R ( M ) is a vonNeumann regular ring [14]. INITE Σ-RICKART MODULES Recall that a module M is said to have D condition if ∀ N ≤ M with M/N ∼ = M ′ ≤ ⊕ M ,we have N ≤ ⊕ M . Note that any Rickart module and any projective module satisfies D condition. Dually, M is said to have C condition if ∀ N ≤ M with N ∼ = M ′ ≤ ⊕ M , wehave N ≤ ⊕ M . Proposition 1.2.
The following statements hold true for a right R -module M : (i) ([12, Proposition 2.11]) M is a Rickart module if and only if M has D conditionand Im ϕ is isomorphic to a direct summand of M for all ϕ ∈ End R ( M ) . (ii) ([14, Theorem 1.11]) M is an endoregular module if and only if M is a Rickartmodule and M has C condition. (iii) ([14, Proposition 2.26]) M is a projective Rickart module if and only if Im ϕ isprojective for each ϕ ∈ End R ( M ) . A module M is said to be N -Rickart (or relatively Rickart to N ) if Ker ρ ≤ ⊕ M for everyhomomorphism ρ ∈ Hom R ( M, N ) [19].
Theorem 1.3 ([13, Theorem 2.6]) . Let M and N be modules. Then M is N -Rickart if andonly if for any direct summand M ′ ≤ ⊕ M and any submodule N ′ ≤ N , M ′ is N ′ -Rickart. In some results we will assume that M has some projectivity conditions in order to getdeeper results. The next lemma will be useful. Lemma 1.4 ([22, 18.2]) . The following statements hold true for a right R -module M : (i) Consider → N ′ → N → N ′′ → as a short exact sequence. If M is an N -projective module then M is N ′ - and N ′′ -projective. (ii) If M is N i -projective for right R -modules N , . . . , N ℓ , then M is L ℓi =1 N i -projective. Recall that a right R -module M is called Σ -Rickart if every direct sum of copies of M is a Rickart module [11]. Also, Add( M ) denotes the class of all right R -modules K suchthat K is isomorphic to a direct summand of M ( I ) for some nonempty index set I (see [11,Definition 2.9]). Proposition 1.5 ([11, Proposition 2.11]) . Let M be a right R -module such that R ∈ Add( M ) . Then M is a Σ -Rickart module if and only if M is a projective R -module and R is a right hereditary ring. Theorem 1.6 ([11, Theorem 4.6]) . The following conditions are equivalent for a finitelygenerated module M : (a) M is a Σ -Rickart module; (b) S = End R ( M ) is a right hereditary ring and S M is flat. Finite Σ -Rickart Modules In this section, after we introduce Σ-Rickart modules in 2020 [11], we present anothernatural generalized notion which is called finite Σ-Rickart modules and obtain some of itsbasic properties. Note that, since proofs of some results are similar to those in [11], we willomit or include proofs for the convenience of the reader.
Definition 2.1.
A right R -module M is called finite Σ -Rickart if every finite direct sumof copies of M is a Rickart module. Example 2.2. (i) R R is a finite Σ-Rickart module iff R is a right semi-hereditary ring.(ii) Any K -nonsingular continuous module is finite Σ-Rickart.(iii) Every Σ-Rickart module and every endoregular module are finite Σ-Rickart. GANGYONG LEE AND MAURICIO MEDINA-B ´ARCENAS (iv) Any submodule of Q Z is a finite Σ-Rickart module. For, let N be any submoduleof Q Z and ϕ : N ( n ) → N ( n ) be any endomorphism for any 0 < n ∈ N . Then Im ϕ is atorsion-free group. Hence Ker ϕ is a pure subgroup of N ( n ) by [5, Ch.V, 26(d)]. ThereforeKer ϕ ≤ ⊕ N ( n ) by [6, Lemma 86.8]. Thus N is a finite Σ-Rickart module.(v) Let R be a Dedekind domain and M be a direct sum of finitely generated torsion-free R -modules of rank one. Then every submodule of M is a finite Σ-Rickart module ([9,Theorems 3 and 4]).(vi) Every finitely generated free (projective) module over a right semi-hereditary ring isa finite Σ-Rickart module.(vii) When M = L i ∈I M i with M i E M for all i ∈ I , L i ∈I M i is a finite Σ-Rickartmodule if and only if M i is a finite Σ-Rickart module for all i ∈ I .We have the implications for right R -modules:(2.1) Σ -Rickart ⇓ Endoregular = ⇒ finite Σ -Rickart = ⇒ Rickart
The next examples show that each converse of the above implications is not true, ingeneral.
Example 2.3. (i) Z [ x ] Z [ x ] is Rickart but it is not finite Σ-Rickart.(ii) The localization of integers at a prime p , Z ( p ) = { ab | a, b ∈ Z , p ∤ b } , is a finite Σ-Rickart Z -module which is not endoregular.(iii) If a module has C condition then the three concepts in the low part of (2.1) coincideby Proposition 1.2(ii).(iv) Consider the Z -module M = Q ⊕ Z . Since Z ( n ) is a nonsingular extending module forany n ∈ N and E ( Z ( n ) ) = Q ( n ) , from [13, Theorem 2.16] M ( n ) is a Rickart module for any n ∈ N . Thus, M is a finite Σ-Rickart module. However, M is not a Σ-Rickart module. For,assume that M is a Σ-Rickart module. Since Z ≤ ⊕ M , by Proposition 1.5 M is a projectivemodule, a contradiction. Lemma 2.4. (i)
Every direct summand of a finite Σ -Rickart module is finite Σ -Rickart. (ii) Every finite direct sum of copies of a finite Σ -Rickart module is finite Σ -Rickart.Proof. (i) Let M be a finite Σ-Rickart module and N be a direct summand of M . Then N ( n ) is a direct summand of M ( n ) for all 0 < n ∈ N . Since M ( n ) is a Rickart module, so is N ( n ) . Thus, N is a finite Σ-Rickart module.(ii) Let M be a finite Σ-Rickart module. Consider M ( n ) a direct sum of copies of M forany n ∈ N . Then ( M ( n ) ) ( m ) = M ( nm ) is a Rickart module for all n, m ∈ N . Therefore M ( n ) is a finite Σ-Rickart module. (cid:3) Definition 2.5.
Let M be a right R -module. Denote by add( M ) the class of all right R -modules K such that K is isomorphic to a direct summand of M ( n ) for some 0 < n ∈ N .Note that add( R ) consists of all finitely generated projective right modules over a ring R . Remark 2.6. If M is a right R -module such that R ∈ add( M ), then M is a projective left S -module where S = End R ( M ). For, since R is in add( M ), M ( n ) ∼ = R ⊕ N for some right R -module N and some n ∈ N . Applying the functor Hom R ( , M ) we get S ( n ) ∼ = Hom R ( R, M ) ⊕ Hom R ( N, M ) ∼ = M ⊕ Hom R ( N, M ) as left S -modules. Thus, S M is projective. In addition,for the case of Add( M ), if M R is finitely generated such that R ∈ Add( M ), then S M isprojective.The next proposition generalizes Lemma 2.4(ii). INITE Σ-RICKART MODULES Proposition 2.7.
A module M is finite Σ -Rickart if and only if every element in add( M ) is a finite Σ -Rickart module. Recall that a module N is said to be finitely M -generated if there exists an epimorphism M ( n ) → N for some 0 < n ∈ N . Lemma 2.8.
For a finite Σ -Rickart module M , the following statements hold true: (i) M ( m ) is M ( n ) -Rickart for every < m, n ∈ N . (ii) For given K ∈ add( M ) , the intersection of two finitely M -generated submodules of K is finitely M -generated. (iii) The intersection of two finitely M -generated submodules of M is finitely M -generated.Proof. (i) It directly follows from Theorem 1.3. (ii) The proof is similar to that of [11,Lemma 2.13]. (iii) It is the special case of (ii) (see also Theorems 3.10 and 3.20). (cid:3) Corollary 2.9 (e.g., [10, Corollary 4.60]) . For a right semi-hereditary ring R , the intersec-tion of two finitely generated ideals of R is finitely generated.Proof. It directly follows from Lemma 2.8(iii) (see also Corollary 3.12 and Lemma 3.19). (cid:3)
Theorem 2.10.
The following conditions are equivalent for a module M : (a) M is a finite Σ -Rickart module; (b) every K ∈ add( M ) satisfies the following two statements: (1) any finitely M -generated submodule of K is in add( M ) ; and (2) any epimorphism N → K with N finitely M -generated splits.Proof. The proof is similar to that of [11, Theorem 2.12]. (cid:3)
The following examples show that Conditions (b1) and (b2) of Theorem 2.10 are inde-pendent.
Example 2.11. (i) Let M Z = Z p ∞ for a prime p ∈ Z and let K ∈ add( M ) be arbitrary.Consider P as a finitely M -generated submodule of K . Then there exists an epimorphism ρ : M ( n ) → P for some n >
0. Since M is divisible, so is M ( n ) . It is a fact that epimorphicimages of divisible groups are divisible, hence P is divisible. This implies that P ≤ ⊕ K .Thus P ∈ add( M ). Therefore M = Z p ∞ satisfies Theorem 2.10(b1).Now, consider the epimorphism ϕ : M → M given by ϕ ( a ) = ap . Since ϕ is not amonomorphism and M is uniform, ϕ does not split. Thus M does not satisfy Theorem2.10(b2). Note that M is not finite Σ-Rickart because M is not a Rickart Z -module.(ii) Consider the ring R = n(cid:16) a ( x,y )0 a (cid:17) (cid:12)(cid:12) a ∈ Z , ( x, y ) ∈ Z ⊕ Z o with the usual addition and multiplication of matrices. Then R is a commutative localartinian ring with maximal ideal I = n(cid:16) x,y )0 0 (cid:17) (cid:12)(cid:12) ( x, y ) ∈ Z ⊕ Z o . Let M be a finitely generated free R -module. Then M satisfies Theorem 2.10(b2) becauseevery element in add( M ) is projective. However, let N be a simple submodule of M . Since M is a free module, N is finitely M -generated. Since R is local, r R ( N ) ≤ ess R . Thus N isa singular simple right R -module. Hence N is not projective, that is, N is not in add( M ).Therefore M does not satisfy Theorem 2.10(b1). Note that R is not a Rickart R -modulebecause Ker (cid:16) , (cid:17) ≤ ess R R , hence M is not finite Σ-Rickart. Theorem 2.12. If M is a finite Σ -Rickart module then every finitely M -generated sub-module P of any element in add( M ) is isomorphic to a direct sum of finitely M -generatedsubmodules of M . GANGYONG LEE AND MAURICIO MEDINA-B ´ARCENAS
Proof.
The proof is similar to that of [11, Theorem 2.14]. (cid:3)
Theorem 2.13.
The following conditions are equivalent for a module M : (a) M is a finite Σ -Rickart module; (b) every finitely M -generated submodule of any element in add( M ) has D condition.Proof. The proof is similar to that of [11, Theorem 2.17]. (cid:3)
Corollary 2.14.
The following conditions are equivalent for a module M : (a) M is a quasi-projective finite Σ -Rickart module; (b) every finitely M -generated submodule of any element in add( M ) is M -projective; (c) every finitely M -generated submodule of any element in add( M ) is quasi-projective.Proof. (a) ⇒ (b) Let K be a finitely M -generated submodule of an element in add( M ). ByTheorem 2.10, K ∈ add( M ), that is, K is isomorphic to a direct summand of M ( n ) for some n >
0. Since M is quasi-projective, M ( n ) is M -projective. Hence K is M -projective.(b) ⇒ (c) Let K be a finitely M -generated submodule of an element in add( M ). Thenthere exists an epimorphism M ( n ) → K for some n >
0. Since K is M -projective, K is K -projective by Lemma 1.4. Therefore K is quasi-projective. (c) ⇒ (a) It follows fromTheorem 2.13. (cid:3) Corollary 2.15.
Let R be a Dedekind domain which is a complete discrete valuation ring.Then every torsion-free module of finite rank is a quasi-projective finite Σ -Rickart module.Proof. Let M be a torsion-free R -module of finite rank. Let N be a finitely M -generatedsubmodule of an element in add( M ). Then N is torsion-free and has finite rank. Then N isquasi-projective by [18, Theorem 5.8]. From Corollary 2.14, M is a quasi-projective finiteΣ-Rickart module. (cid:3) It is well known that a ring R is right semi-hereditary if and only if every finitely generatedsubmodule of a right projective module is projective ([2, Proposition 6.2]). In the next result,we give more characterizations for right semi-hereditary rings. Corollary 2.16.
The following conditions are equivalent for a ring R : (a) R is a right semi-hereditary ring; (b) every finitely generated submodule of any projective right R -module is projective; (c) every finitely generated submodule of any projective right R -module is R -projective; (d) every finitely generated submodule of any projective right R -module is quasi-projective; (e) every finitely generated submodule of any projective right module has D condition. In [11] Σ-Rickart modules were characterized using a class of modules called E M . For aright R -module M , it is denoted by E M the class of all right R -modules A such that for anymonomorphism α : N → M with N an M -generated module and for any homomorphism β : N → A , there exists γ : M → A such that β = γα . For the analogue of the above classrelated to finite Σ-Rickart modules, we introduce the following. Definition 2.17.
Let M be a right R -module. Denote by F M the class of all right R -modules A such that for any monomorphism α : N → M with N a finitely M -generatedmodule and for any homomorphism β : N → A , there exists γ : M → A such that β = γα .For a right R -module M , a right R -module A is said to be f M -injective if for any finitely M -generated submodule N of M and for any homomorphism β : N → A , there exists ahomomorphism γ : M → A such that γ | N = β . Note that in the case of M = R R , A is saidto be f-injective if A is f R -injective (see [8]). We can easily see that the every element in F M is exactly f M -injective as the following. INITE Σ-RICKART MODULES Proposition 2.18.
For a right R -module M , a module A is in F M iff A is f M -injective. Proposition 2.19.
For a right R -module M , a module A is in F M if and only if for anymonomorphism α : N → K with N a finitely M -generated module and K ∈ add( M ) , andfor any homomorphism β : N → A , there exists γ : K → A such that β = γα .Proof. The proof is similar to that of [11, Proposition 3.2]. (cid:3)
Remark 2.20. (i) We have the following contentions, E R ⊆ { all M -injective modules } ⊆ E M ⊆ F M = { all f M -injective modules } where E R = { all injective modules } . Note that F R = { all f-injective modules } .(ii) If every submodule of M is finitely M -generated then every module in F M is M -injective. Proposition 2.21.
The following statements hold true for a right R -module M : (i) F M is closed under direct products. (ii) F M is closed under direct summands. (iii) If M is in F M then M has C condition. (iv) If every finitely M -generated submodule of A is in F M , then A is in F M .Proof. All proofs are similar to those of [11, Proposition 3.6]. However, we give the proofof (iv) for the convenience of the reader. (iv) Let α : N → M be a monomorphism with N finitely M -generated and let β : N → A be any homomorphism. Since N is finitely M -generated, Im β is finitely M -generated. Because Im β ⊆ A , by hypothesis there exists γ : M → Im β such that β ( N ) = γα ( N ). Therefore A ∈ F M . (cid:3) Corollary 2.22.
If every finitely generated submodule of M is f-injective then M is alsof-injective. Proposition 2.23.
The following conditions are equivalent for a module M : (a) M is an endoregular module; (b) M has D condition and F M = Mod - R .Proof. (a) ⇒ (b) It is clear that M has D condition. Let L be any right R -module and let N a finitely M -generated submodule of M . Then there exists an epimorphism ρ : M ( n ) → N for some n >
0. Also, let α : N → M be any monomorphism and β : N → L beany homomorphism. By [14, Corollary 3.15], M ( n ) is an endoregular module, and hence αρ ( M ( n ) ) = α ( N ) is a direct summand of M . Take γ = βα − ⊕
0. Then γ : M → L is ahomomorphism such that γα = β . Therefore L is in F M .(b) ⇒ (a) Let ϕ : M → M be any endomorphism of M . Then Im ϕ is finitely M -generated.Since Im ϕ is in F M , the canonical inclusion j : Im ϕ → M splits, that is, Im ϕ is a directsummand of M . By the D condition, we can infer that Ker ϕ is a direct summand of M .Thus, M is an endoregular module. (cid:3) An epimorphism µ : A → B is called an M -pure epimorphism if for any homomorphism β : M → B , there exists γ : M → A such that µγ = β [22] (see also [11, Proposition 3.10]). Remark 2.24.
It is not difficult to see that:(i) An epimorphism µ : A → B is an M -pure epimorphism if and only if µ is a K -pureepimorphism for any K in add( M ).(ii) For a projective module M , every epimorphism is an M -pure epimorphism.(iii) If µ : A → B and ν : C → D are M -pure epimorphisms, then µ ⊕ ν : A ⊕ C → B ⊕ D is also an M -pure epimorphism. GANGYONG LEE AND MAURICIO MEDINA-B ´ARCENAS
Lemma 2.25 ([11, Lemma 3.11]) . Let M be M ( I ) -projective for any ( resp., finite ) indexset I . If A is an ( resp., finitely ) M -generated module then every epimorphism µ : A → B is an M -pure epimorphism. This results is a module theoretic version of [15, Theorem 2].
Theorem 2.26.
Consider the following conditions for a module M : (i) M is a finite Σ -Rickart module. (ii) F M is closed under M -pure epimorphisms. (iii) If µ ∈ Hom R ( A, A ′ ) is an M -pure epimorphism with A M -injective, then A ′ ∈ F M .Then the implications (i) ⇒ (ii) ⇒ (iii) hold true. In addition, if M is M ( I ) -projective for anyindex set I , then the three conditions are equivalent.Proof. The proofs are similar to those of [11, Theorem 3.12]. (cid:3)
Remark from [11, Example 3.13] that the converse of (i) ⇒ (ii) is not true, in general.Recall that a submodule N of a right R -module M is said to be pure if for every left R -module K , the canonical homomorphism ι ⊗ N ⊗ R K → M ⊗ R K is a monomorphism,where ι : N → M is the canonical inclusion. M is said to be absolutely pure if M is a puresubmodule of any module which contains M as a submodule (see [15]). Lemma 2.27 ([15, Corollary 2] and Remark 2.20(i)) . A right R -module M is absolutelypure if and only if M is f-injective if and only if M is in F R . As a corollary, we have a characterization for right semi-hereditary rings including [15,Theorem 2].
Corollary 2.28.
The following conditions are equivalent for a ring R : (a) R is a right semi-hereditary ring; (b) Every factor module of any f-injective R -module is f-injective; (c) Every factor module of any absolutely pure R -module is absolutely pure; (d) Every factor module of any injective R -module is absolutely pure. Now, we are going to give a module theoretic version of [8, Theorem 3.4].
Theorem 2.29.
The following conditions are equivalent for a right R -module M : (a) M is an endoregular module; (b) M is a finite Σ -Rickart module and M is in F M ; (c) M is strongly D condition ( i.e., M ( n ) has D condition for all n > and anyfinitely M -generated submodule of M ( n ) is a direct summand for all n > .Proof. (a) ⇔ (b) Let M be an endoregular module. Then by [14, Corollary 3.15], M ( n ) is anendoregular module, which is Rickart. Thus, M is finite Σ-Rickart. Also, from Proposition2.23 M is in F M . Conversely, let M ∈ F M . Then M has C condition from Proposition2.21(iii). Hence M is an endoregular module by [12, Theorem 3.17].(a) ⇒ (c) Since each endoregular module is finite Σ-Rickart, from Theorem 2.13 it is easyto see that M is strongly D condition. Now, let N be a finitely M -generated submodule of M ( n ) for some n >
0. Then there exists an epimorphism ρ : M ( ℓ ) → N for some ℓ >
0. Onthe other hand, let j : N → M ( n ) be the canonical inclusion. Then there is a homomorphism jρ : M ( ℓ ) → M ( n ) . Therefore Im jρ = N is a direct summand of M ( n ) .(c) ⇒ (b) Let ϕ : M ( n ) → M ( n ) be any endomorphism. By hypothesis Im ϕ ≤ ⊕ M ( n ) .Since M ( n ) has D condition, Ker ϕ ≤ ⊕ M ( n ) . Thus M is a finite Σ-Rickart module. Inaddition, let N be a finitely M -generated submodule of M and let β : N → M be anyhomomorphism. By hypothesis, N is a direct summand of M . This implies that β can beextended to a homomorphism γ : M → M . Thus, M is in F M . (cid:3) INITE Σ-RICKART MODULES Corollary 2.30.
The following conditions are equivalent for a ring R : (a) R is a von Neumann regular ring; (b) R is a right semi-hereditary ring and R R is an f -injective module; (c) R is a right semi-hereditary ring and R R is an absolutely pure module; (d) every finitely generated submodule of R ( n ) is a direct summand for all n > . M -coherent modules and the endomorphism ring of a finite Σ -Rickartmodule The next result can be seen as a generalization of Schanuel’s Lemma [10, 5.1].
Lemma 3.1.
Let M be a right R -module and K ∈ add( M ) . Let −→ D σ −→ K ρ −→ C −→ −→ A α −→ B β −→ C −→ be short exact sequences with β an M -pure epimorphism. Then there exists a short exactsequence −→ D δ −→ K ⊕ A η −→ B −→ . Moreover, if ρ is also an M -pure epimorphism, then so is η .Proof. Consider the following diagram:0 / / D σ / / K ρ / / γ (cid:15) (cid:15) ✤✤✤ C / / = (cid:15) (cid:15) / / A α / / B β / / C / / . Since K ∈ add( M ) and β is an M -pure epimorphism, there exists γ : K → B such that ρ = βγ . We claim that the following sequence is exact(3.1) 0 / / D δ / / K ⊕ Im α η / / B / / δ ( d ) = ( σ ( d ) , γ ( σ ( d ))) and η ( k, x ) = γ ( k ) − x for d ∈ D , k ∈ K and x ∈ Im α : It isclear that γ ( σ ( d )) ∈ Im α , δ is a monomorphism, and ηδ = 0. Let ( k, x ) ∈ K ⊕ Im α suchthat η ( k, x ) = 0. That is, γ ( k ) = x . Then 0 = β ( x ) = β ( γ ( k )) = ρ ( k ). Since Im σ = Ker ρ there exists d ∈ D such that σ ( d ) = k . Thus δ ( d ) = ( σ ( d ) , γ ( σ ( d ))) = ( k, γ ( k )) = ( k, x ).Hence Im δ = Ker η . Now, it remains to show that η is an epimorphism. Let b ∈ B . Since ρ is an epimorphism, there exists ℓ ∈ K such that ρ ( ℓ ) = β ( b ). Hence β ( γ ( ℓ ) − b ) = 0, thatis, there exists a ∈ A such that α ( a ) = γ ( ℓ ) − b . Thus η ( ℓ, α ( a )) = b , proving the claim.Since A ∼ = Im α , we have an exact sequence0 → D → K ⊕ A → B → . Moreover, suppose ρ is also an M -pure epimorphism. Let ζ : M → B be any homomor-phism. Then, βζ : M → C . Since ρ is an M -pure epimorphism, there exist ϕ : M → K such that ρϕ = βζ . This implies that βγϕ = βζ and so γϕ ( m ) − ζ ( m ) ∈ Ker β = Im α forall m ∈ M . Define ϕ : M → K ⊕ Im α as ϕ ( m ) = ( ϕ ( m ) , γϕ ( m ) − ζ ( m )). It is clear that ηϕ = ζ . (cid:3) Lemma 3.2.
Let M be a right R -module and −→ A α −→ B β −→ C −→ be an exact sequence with A and C finitely M -generated modules. If β is an M -pure epi-morphism then B is finitely M -generated. Proof.
Since C is finitely M -generated, there exists an epimorphism ρ : M ( n ) → C for some n ∈ N . Consider the pull-back P of ( β, ρ ), that is, P = { ( b, m ) ∈ B ⊕ M ( n ) | β ( b ) = ρ ( m ) } : P ρ ′ (cid:15) (cid:15) β ′ / / M ( n ) ρ (cid:15) (cid:15) / / A α / / B β / / C / / . Note that both ρ ′ and β ′ are epimorphisms. On the other hand, Ker β ′ = Ker β ⊕ ∼ = A .Since β is an M -pure epimorphism, there exists κ : M ( n ) → B . Therefore, β ′ splits.Hence P ∼ = A ⊕ M ( n ) . This implies that P is finitely M -generated because A is finitely M -generated. Since ρ ′ is an epimorphism, B is finitely M -generated. (cid:3) Proposition 3.3.
Let C be a module such that there exists an exact sequence → D → M ( n ) π → C → with π an M -pure epimorphism and D finitely M -generated. If ρ : B → C is an M -pure epimorphism with B finitely M -generated, then Ker ρ is finitely M -generated.Proof. Consider the exact sequence0 → D → M ( n ) π → C → D finitely M -generated. Since ρ and π are M -pure epimorphisms, by Lemma 3.1 weget an exact sequence 0 → D → M ( n ) ⊕ Ker ρ η → B → η an M -pure epimorphism. From Lemma 3.2, Ker ρ is finitely M -generated. (cid:3) Recall that a right R -module M is said to be intrinsically projective if for every diagram M β (cid:15) (cid:15) γ | | ② ② ② ② M ( n ) α / / N / / n > N ≤ M , there exists γ : M → M ( n ) such that αγ = β (see [23]). Note thatevery finite Σ-Rickart module and every quasi-projective module is intrinsically projective.In addition, a right R -module M is intrinsically projective if and only if I = Hom R ( M, IM )for all finitely generated right ideals I ≤ End R ( M ) ([23, 5.7]). Lemma 3.4.
Let M be an intrinsically projective module. Then the following statementshold true: (i) Any epimorphism ρ : L → C , with C ≤ M and L finitely M -generated, is an M -pureepimorphism. (ii) For any finitely M -generated submodule N of M ( n ) with any n ∈ N , Hom R ( M, N ) is a finitely generated right S -module where S = End R ( M ) .Proof. (i) Let ρ : L → C be any epimorphism with C ≤ M and L finitely M -generated.Let α : M → C be any homomorphism. Since L is finitely M -generated there exists anepimorphism β : M ( n ) → L for some n >
0. Since M is intrinsically projective, there exists γ : M → M ( n ) such that α = ( ρβ ) γ = ρ ( βγ ).(ii) Let N ≤ M ( n ) be finitely M -generated. Hence there exist an integer k > ρ : M ( k ) → N . Let ℓ = max { k, n } , then we can see ρ : M ( ℓ ) → N and N ≤ M ( ℓ ) . Let π i : M ( ℓ ) → M denote the canonical projection for each 1 ≤ i ≤ ℓ . Let ϕ : M → N be any homomorphism and consider the epimorphisms π i ρ : M ( ℓ ) → π i ( N )for 1 ≤ i ≤ ℓ . Since M is intrinsically projective there exists γ i : M → M ( ℓ ) such that INITE Σ-RICKART MODULES π i ργ i = π i ϕ for all 1 ≤ i ≤ ℓ . Define γ : M → M ( ℓ ) as γ ( m ) = ( γ ( m ) , . . . , γ ℓ ( m )). Hence ργ = ϕ . Let η i : M → M ( ℓ ) denote the canonical inclusion for each 1 ≤ i ≤ ℓ . Then ϕ = ργ = ℓ X i =1 ( ρη i ) π i γ for π i γ ∈ S. Thus, Hom R ( M, N ) is generated by ρη , ρη , . . . , ρη ℓ . (cid:3) For a right R -module M , a right R -module N is said to be finitely M - presented if thereexists an exact sequence M ( ℓ ) → M ( n ) → N → n, ℓ > Lemma 3.5.
Let M be an intrinsically projective module and C be a finitely M -presentedsubmodule of M . If ρ : B → C is an epimorphism with B finitely M -generated, then Ker ρ is finitely M -generated.Proof. Since C is finitely M -presented there exists an exact sequence0 → D → M ( n ) π → C → D finitely M -generated. Note that ρ and π are M -pure epimorphisms from Lemma3.4(i). Therefore, the result follows from Proposition 3.3. (cid:3) Proposition 3.6.
Let M be an intrinsically projective module and A, B ≤ M be finitely M -presented submodules. Consider the following exact sequence −→ A ∩ B −→ A ⊕ B π −→ A + B −→ . Then A + B is finitely M -presented if and only if A ∩ B is finitely M -generated.Proof. Let A and B be finitely M -presented submodules of M . Then there exist epimor-phisms ρ : M ( n ) → A and ρ : M ( n ) → B for some n , n ∈ N . So, ρ = ρ ⊕ ρ : M ( n ) ⊕ M ( n ) → A ⊕ B is an epimorphism. That is, A ⊕ B is finitely M -generated.Suppose A + B is finitely M -presented. Since M is intrinsically projective and A ⊕ B isfinitely M -generated, A ∩ B is finitely M -generated by Lemma 3.5.Conversely, since A and B are finitely M -presented, there is an exact sequence0 → Ker ρ → M ( n ) ⊕ M ( n ) ρ → A ⊕ B → ρ finitely M -generated and ρ = ρ ⊕ ρ an M -pure epimorphism by Lemma 3.4(i)and Remark 2.24(iii). Consider the exact sequence0 → Ker πρ → M ( n + n ) πρ → A + B → . Note that π is an M -pure epimorphism by Lemma 3.4(i) because A ⊕ B is finitely M -generated. Hence from Lemma 3.1, we have an exact sequence0 → Ker πρ → ( A ∩ B ) ⊕ M ( n + n ) η → A ⊕ B → η an M -pure epimorphism. Since A ∩ B is finitely M -generated, ( A ∩ B ) ⊕ M ( n + n ) is finitely M -generated. Also since Ker ρ is finitely M -generated, Ker η ∼ = Ker πρ is finitely M -generated by Proposition 3.3. This implies that A + B is finitely M -presented. (cid:3) Definition 3.7.
Let M be a right R -module and N be a finitely M -generated module. Themodule N is called M - coherent if for any n > ρ : M ( n ) → N ,Ker ρ is finitely M -generated.Remark that if a right R -module M is M -coherent then S M is flat where S = End R ( M ).Also, a ring R is said to be right coherent if R R is R -coherent. In addition, M is a coherentright R -module if and only if M is an R -coherent right R -module ([10, 4G]). Lemma 3.8.
Let M be an M -coherent module. If ρ : B → C is an epimorphism with B finitely M -generated and C ≤ M , then Ker ρ is finitely M -generated.Proof. Let ρ : B → C be an epimorphism with B finitely M -generated and C ≤ M . Hencethere exists an epimorphism π : M ( n ) → B for some n >
0. Since M is M -coherent,Ker ρπ = π − (Ker ρ ) is finitely M -generated. Since Ker ρ is a factor module of Ker ρπ ,Ker ρ is finitely M -generated. (cid:3) Proposition 3.9. M is an M -coherent module if and only if every finitely M -generatedsubmodule of M is finitely M -presented.Proof. It directly follows from Lemma 3.8 and the definition of an M -coherent module. (cid:3) Theorem 3.10.
Consider the following conditions for a module M : (i) M is an M -coherent module. (ii) The intersection of two finitely M -generated submodules of M is finitely M -generatedand Ker ϕ is finitely M -generated for all ϕ ∈ End R ( M ) .Then (i) ⇒ (ii) holds. In addition, if M is intrinsically projective then the two conditionsare equivalent.Proof. (i) ⇒ (ii) By the definition of an M -coherent module, Ker ϕ is finitely M -generatedfor every ϕ ∈ End R ( M ). Now, let A and B be finitely M -generated submodules of M .Hence A ⊕ B is finitely M -generated. Consider the natural exact sequence0 → A ∩ B → A ⊕ B → A + B → . It follows from Lemma 3.8 that A ∩ B is finitely M -generated.In addition, suppose M is intrinsically projective. For (ii) ⇒ (i), we are going to prove byinduction on n that the kernel of any homomorphism ϕ : M ( n ) → M is finitely M -generated.If n = 1, the kernel of any ϕ ∈ End R ( M ) is finitely M -generated by hypothesis. Suppose n > ρ : M ( ℓ ) → M with ℓ < n , Ker ρ is finitely M -generated.Let ψ : M ( n ) → M be any homomorphism. We have that Im ψ = ψ (0 ⊕ M ( n − ) + ψ ( M ⊕ ψ | ⊕ M ( n − ) is finitely M -generated by the induction hypothesis andKer( ψ | M ⊕ ) is finitely M -generated, ψ (0 ⊕ M ( n − ) and ψ ( M ⊕
0) are finitely M -presented.By hypothesis, ψ (0 ⊕ M ( n − ) ∩ ψ ( M ⊕
0) is finitely M -generated. It follows from Proposition3.6 that Im ψ = ψ (0 ⊕ M ( n − ) + ψ ( M ⊕
0) is finitely M -presented. Hence Ker ψ is finitely M -generated by Lemma 3.5. Thus, M is an M -coherent module. (cid:3) Corollary 3.11 ([10, Corollary 4.60]) . A ring R is a right coherent ring if and only if theintersection of two finitely generated ideals of R is finitely generated and r R ( a ) is finitelygenerated for all a ∈ R . Corollary 3.12.
Let M be an intrinsically projective Rickart module. Then M is M -coherent if and only if the intersection of two finitely M -generated submodules of M isfinitely M -generated. Chase ([10, 4.60]) shows that a domain R is right coherent if and only if the intersectionof two finitely generated right ideals of R is finitely generated. In the next result, we extendto a right Rickart ring. Corollary 3.13.
A right Rickart ring R is right coherent if and only if the intersection oftwo finitely generated right ideals of R is finitely generated. Lemma 3.14 ([22, 15.9]) . The following conditions are equivalent for a module M : (a) S M is flat where S = End R ( M ) ; (b) for any homomorphism ρ : M ( n ) → M ( k ) with n, k > , Ker ρ is M -generated; INITE Σ-RICKART MODULES (c) for any homomorphism ρ : M ( n ) → M with n > , Ker ρ is M -generated. Note that if a module N is M -generated then N = Hom R ( M, N ) M . For an intrinsicallyprojective module M , there is another characterization when M is M -coherent as well asTheorem 3.10. Theorem 3.15.
Consider the following conditions for a module M : (i) S is a right coherent ring and S M is flat. (ii) M is an M -coherent module.Then (i) ⇒ (ii) holds. In addition, if M is intrinsically projective then the two conditionsare equivalent.Proof. (i) ⇒ (ii) Let N ≤ M be finitely M -generated. Consider the exact sequence 0 → K → M ( n ) ρ → N →
0. Applying Hom R ( M, ), we get0 → Hom R ( M, K ) → Hom R ( M, M ( n ) ) ρ ∗ → Hom R ( M, N ) . Since N ≤ M , Hom R ( M, N ) embeds in S . Note that Im ρ ∗ is finitely generated as a right S -module. This implies that Hom R ( M, K ) is finitely generated as a right S -module because S is a right coherent ring. Hence there exists an epimorphism S ( ℓ ) → Hom R ( M, K ) for some ℓ >
0. Note that K is M -generated because S M is flat from Lemma 3.14. Applying ⊗ S M , S ( ℓ ) ⊗ S M / / ∼ = (cid:15) (cid:15) Hom R ( M, K ) ⊗ S M / / ∼ = (cid:15) (cid:15) M ( ℓ ) / / K / / . Thus K is finitely M -generated. This implies that M is M -coherent.In addition, suppose M is intrinsically projective. For (ii) ⇒ (i), it is easy to see that S M is flat from Lemma 3.14. Let I be a finitely generated right ideal of S . Then there is anexact sequence 0 → Ker η → S ( n ) η → I → n >
0. It is enough to show that J = Ker η is finitely generated as a right S -module. Applying the functor ⊗ S M , since S M is flat we get(3.2) 0 / / J ⊗ S M / / ∼ = (cid:15) (cid:15) S ( n ) ⊗ S M η ⊗ / / α ∼ = (cid:15) (cid:15) I ⊗ S M / / β ∼ = (cid:15) (cid:15) / / J ′ / / M ( n ) / / IM / / , where α, β are the canonical homomorphisms and J ′ = Ker β ( η ⊗ α − . Since M isintrinsically projective and IM ≤ M , the functor Hom R ( M, ) is exact in (3.2). Therefore,the following diagram has exact rows:0 / / J (cid:15) (cid:15) / / S ( n ) ∼ = (cid:15) (cid:15) η / / I ∼ = (cid:15) (cid:15) / / / / Hom R ( M, J ⊗ S M ) ∼ = (cid:15) (cid:15) / / Hom R ( M, S ( n ) ⊗ S M ) ∼ = (cid:15) (cid:15) / / Hom R ( M, I ⊗ S M ) ∼ = (cid:15) (cid:15) / / / / Hom R ( M, J ′ ) / / Hom R ( M, M ( n ) ) / / Hom R ( M, IM ) / / . Hence J ∼ = Hom R ( M, J ′ ). Since M is M -coherent, J ′ is finitely M -generated. Therefore J is a finitely generated right S -module by Lemma 3.4(ii). (cid:3) Corollary 3.16.
The following are equivalent for an intrinsically projective module M : (a) M is an M -coherent module; (b) S = End R ( M ) is a right coherent ring and S M is flat; (c) The intersection of two finitely M -generated submodules of M is finitely M -generatedand Ker ϕ is finitely M -generated for all ϕ ∈ End R ( M ) . Proposition 3.17.
Let M be a finite Σ -Rickart module. Then the following statementshold true: (i) End R ( M ) is a right semi-hereditary ring. (ii) End R ( M ) is a right coherent ring. (iii) Every finitely M -generated submodule of M is M -coherent.Proof. (i) Since M ( n ) is Rickart, Mat n ( S ) is a right Rickart ring for all 0 < n ∈ N with S = End R ( M ). Then S is a right semi-hereditary ring by [21, Proposition]. (ii) It is trivial(see Lemma 3.19).(iii) Let N be a finitely M -generated submodule of M and let ϕ : M ( n ) → N be anyhomomorphism. Since N ≤ M and M is finite Σ-Rickart, Ker ϕ ≤ ⊕ M ( n ) . Hence Ker ϕ isfinitely M -generated. (cid:3) Lemma 3.18 ([22, 39.10(2)]) . If S = End R ( M ) is a right Rickart ring then r S ( ϕ ) M ≤ ⊕ M for all ϕ ∈ S . Lemma 3.19 (Chase [3, Theorem 4.1]) . A ring R is right semi-hereditary if and only if R is a right coherent ring and all right ideals of R are flat. As a finitely generated Σ-Rickart module is characterized in terms of its endomorphismring (Theorem 1.6), we obtain the characterization of a finite Σ-Rickart module using itsendomorphism ring.
Theorem 3.20.
The following conditions are equivalent for a module M and S = End R ( M ) : (a) M is a finite Σ -Rickart module; (b) S is a right semi-hereditary ring and S M is flat; (c) M is an intrinsically projective M -coherent module and all right S -ideals are flat.Proof. (a) ⇒ (b) It follows from Proposition 3.17(i) and Lemma 3.14.(b) ⇒ (c) Since S is a right semi-hereditary ring, S is a right coherent ring and every right S -ideal is flat by Lemma 3.19. It follows from [23, Examples 5.6(2)] that M is intrinsicallyprojective. From Theorem 3.15 M is M -coherent because S M is flat.(c) ⇒ (a) Since M is intrinsically projective and M -coherent, by Theorem 3.15 S is a rightcoherent ring and S M is flat. From Lemma 3.19 S is a right semi-hereditary ring becauseall right S -ideals are flat. Let ϕ : M ( n ) → M ( n ) be any endomorphism. Since S M is flat asabove, Ker ϕ is M -generated by Lemma 3.14. Hence Ker ϕ = Hom R ( M ( n ) , Ker ϕ ) M ( n ) = r Mat n ( S ) ( ϕ ) M ( n ) ≤ ⊕ M ( n ) from Lemma 3.18. Therefore M is a finite Σ-Rickart module. (cid:3) The “ S M is flat” condition in (b) ⇒ (a) is not superfluous as shown next. Example 3.21. (i) Consider Z p ∞ as a Z -module. Then S = End Z ( Z p ∞ ) is a right semi-hereditary ring. But Z p ∞ is neither a finite Σ-Rickart Z -module nor a flat left S -module.(ii) The Z -module Z is Z -coherent, however Z is not finite Σ-Rickart.An explicit application of Theorem 3.20 is exhibited in the next example. INITE Σ-RICKART MODULES Example 3.22.
Let R be the ring of n × n upper triangular matrices over a right semi-hereditary ring A . Let e ∈ R be a unit matrix with 1 in the (1 , R ( eR ) ∼ = A and eR ∼ = A ( n ) as projective left A -modules. Therefore eR is a finiteΣ-Rickart module by Theorem 3.20. For example, while R = (cid:0) Z Z Z (cid:1) is not a right hereditaryring, eR = (cid:0) Z Z (cid:1) is a finite Σ-Rickart R -module for e = ( ).Since every finitely generated projective module over a right semi-hereditary ring is afinite Σ-Rickart module, its endomorphism ring is a right semi-hereditary ring as a conse-quence of Theorem 3.20. Corollary 3.23.
The following statements hold true: (i) ([4, Theorem 2.10]) If R is a right semi-hereditary ring and P is a finitely generatedprojective R -module, then End R ( P ) is a right semi-hereditary ring. (ii) If R is a right semi-hereditary ring, so is eRe for any idempotent e ∈ R . Applications
Proposition 4.1.
Let M be a right R -module with S = End R ( M ) such that S M is flat.Then the following equivalences hold true: (i) A right R -module A is in E M iff Hom R ( M, A ) is an injective right S -module. (ii) A right R -module A is in F M iff Hom R ( M, A ) is an f-injective right S -module.Proof. (i) Suppose A ∈ E M . Let I S be a right ideal of S and let α : I → Hom R ( M, A ) beany S -homomorphism. Hence we have the following diagram of right R -modules0 / / I ⊗ S M ι ⊗ / / α ⊗ (cid:15) (cid:15) S ⊗ S M ∼ = M g { { ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ Hom R ( M, A ) ⊗ S M j (cid:15) (cid:15) A where ι : I S → S is the canonical inclusion and j : Hom R ( M, A ) ⊗ S M → A is given by j ( f ⊗ m ) = f ( m ). Note that I ⊗ S M is M -generated and since S M is flat, ι ⊗ θ denote the canonical isomorphism S ⊗ S M → M . By the definitionof E M , there exists an R -homomorphism g : M → A such that gθ ( ι ⊗
1) = j ( α ⊗ α : S → Hom R ( M, A ) as ( ¯ α ( f ))( m ) = gf ( m ) for f ∈ S . Let h ∈ I and m ∈ M . Hence( ¯ αι ( h ))( m ) = gh ( m ) = g ( θ ( h ⊗ m )) = ( gθ ( ι ⊗ h ⊗ m )= j ( α ⊗ h ⊗ m ) = j ( α ( h ) ⊗ m ) = ( α ( h ))( m ) . This implies that ¯ αι ( h ) = α ( h ) for all h ∈ I . Thus, Hom R ( M, A ) is an injective right S -module.Conversely, let A R be an R -module such that Hom R ( M, A ) is an injective right S -module.Let N be an M -generated submodule of M and f : N → A be any R -homomorphism. Hencewe have the following diagram of right S -modules0 / / Hom R ( M, N ) i ∗ / / f ∗ (cid:15) (cid:15) S α y y s s s s s s Hom R ( M, A ) where i : N → M is the canonical inclusion. By hypothesis there exists an S -homomorphism α : S → Hom R ( M, A ) such that αi ∗ = f ∗ . Define ¯ α : M → A as ¯ α ( m ) = ( α (Id M )) ( m ). Let n ∈ N be arbitrary. Since N is M -generated, n = P ki =1 g i ( m i ) with g i ∈ Hom R ( M, N ) and m i ∈ M . Then¯ αi ( n ) = ( α (Id M )) ( i ( n )) = ( α (Id M )) i k X i =1 g i ( m i ) ! = k X i =1 ( α (Id M )) ig i ( m i ) = k X i =1 ( α ( ig i )) ( m i )= k X i =1 ( αi ∗ ( g i )) ( m i ) = k X i =1 ( f ∗ ( g i )) ( m i ) = k X i =1 f g i ( m i ) = f k X i =1 g i ( m i ) ! = f ( n )because g i ∈ S . This implies that f = ¯ αi . Thus, A R is in E M .(ii) The proof is similar to that of (i). Note that if I S is a finitely generated ideal of S , then I ⊗ S M is a finitely M -generated right R -module. (cid:3) Remark 4.2. (i) For any endoregular module M , Hom R ( M, A ) is an f-injective right S -module for any module A .(ii) ([15, Theorem 5 and Corollary 2]) Every module over a von Neumann regular ring isf-injective. Corollary 4.3.
Let M be a right R -module with S = End R ( M ) such that S M is flat. Thenthe following equivalences hold true: (i) M ∈ E M if and only if S is a right self-injective ring. (ii) M ∈ F M if and only if S is a right f-injective ring. Here we have an alternative proof of Theorem 2.29.
Corollary 4.4.
The following conditions are equivalent for a right R -module M : (a) M is a finite Σ -Rickart module and M ∈ F M ; (b) End R ( M ) is a von Neumann regular ring.Proof. The proof follows from Theorems 2.29 and 3.20 and Corollaries 2.30 and 4.3(ii). (cid:3)
Corollary 4.5.
The following conditions are equivalent for a finitely generated module M : (a) M is a Σ -Rickart module and M ∈ E M ; (b) End R ( M ) is semisimple artinian.Proof. (a) ⇒ (b) By [11, Theorem 4.6] S = End R ( M ) is a right hereditary ring and S M isflat. It follows from Corollary 4.3(i) that S is right self-injective. Thus, S S is semisimpleartinian by [17, Corollary]. (b) ⇒ (a) Since S is von Neumann regular, from Theorems 2.29and 3.20 S M is flat. So, the proof follows from [11, Theorem 4.6] that M is Σ-Rickart andfrom Corollary 4.3(i) that M ∈ E M . (cid:3) Continuing the study of the endomorphism ring of a finite Σ-Rickart module, we studythe case when End R ( M ) is a semiprimary ring (Theorem 4.6). Recall that a ring R is saidto be semiprimary if its Jacobson radical, Rad R , is nilpotent and R/ Rad R is a semisimpleartinian ring.Recall that a ring R is called a PWD ( piecewise domain ) if it possesses a complete set { e , ..., e n } of orthogonal idempotents such that xy = 0 implies x = 0 or y = 0 whenever x ∈ e i Re k and y ∈ e k Re j (see [7]). Theorem 4.6.
The following conditions are equivalent for a module M : (a) M has a decomposition L ni =1 H ( ℓ i ) i with H i an indecomposable endoregular module, H i is H j -Rickart for all ≤ i, j ≤ n , and H i ≇ H j for i = j ; INITE Σ-RICKART MODULES (b) S = End R ( M ) is isomorphic to a upper triangular matrix ring Mat ℓ ( D ) V V · · · V n ℓ ( D ) V · · · V n ℓ ( D ) · · · V n ... ... ... . . . ... · · · Mat ℓ n ( D n ) where D i is a division ring for ≤ i ≤ n , and V ij is a Mat ℓ i ( D i ) - Mat ℓ j ( D j ) -bimodulefor all ≤ i < j ≤ n satisfying l S ( x ) ∩ V ij = 0 for any = x ∈ H j . In particular, S is a semiprimary PWD.Proof. (a) ⇒ (b) Since each H i is indecomposable endoregular and H i is H j -Rickart module,every nonzero homomorphism ρ : H i → H j is a monomorphism. This implies that S is aPWD [7]. On the other hand, since H i ≇ H j for all 1 ≤ i = j ≤ n , Hom R ( H i , H j ) = 0 orHom R ( H j , H i ) = 0 from [24, Proposition 18]. Without loss of generality, we can assumethat the decomposition M = L ni =1 H ( ℓ i ) i is such that Hom R ( H i , H j ) = 0 for all i < j .Consider the complete set of orthogonal primitive idempotents { e , . . . , e n } of S such that H ( ℓ i ) i = e i M . It follows from [7, Main Theorem] that S = P V · · · V n P · · · V n ... ... . . . ...0 0 · · · P n where V ij is a P i - P j -bimodule and P i = D i W · · · W ℓ i W D i · · · W ℓ i ... ... . . . ... W ℓ i W ℓ i · · · D i with D i a division ring and each W jk ∼ = D i as D i - D i -bimodule. That is, P i ∼ = Mat ℓ i ( D i ).Suppose that Hom R ( H j , H i ) = 0 with 1 ≤ i < j ≤ n . It follows from [13, Corollary 2.10]that H j is H ( ℓ i ) i -Rickart. Therefore, every nonzero homomorphism in Hom R ( H j , H ( ℓ i ) i ) isa monomorphism. Let 0 = x ∈ H j and ϕ ∈ S such that ϕ ∈ l S ( x ) ∩ V ij . Then ϕ ( x ) = 0.If V ij = 0, there is nothing to prove. Suppose that V ij = Hom R ( H ( ℓ j ) j , H ( ℓ i ) i ) = 0. Assumethat ϕ = 0. Since ϕ | H j is a monomorphism by the above comment, x = 0, a contradiction.Hence ϕ = 0. Therefore l S ( x ) ∩ V ij = 0. Note that Rad( S ) = V ··· V n ··· V n ... ... ... ... ··· ! is nilpotentand S/ Rad( S ) ∼ = Mat ℓ ( D ) × · · · × Mat ℓ n ( D n ) which is a semisimple artinian ring. Hence S is a semiprimary ring.(b) ⇒ (a) Let { e ij | ≤ i, j ≤ m } denote the matrix units where m = ℓ + · · · + ℓ n .Hence M has a decomposition M = e M ⊕ · · · ⊕ e mm M . Denote H i = e ii M . SinceEnd R ( H i ) ∼ = e ii Se ii ∼ = D i is a division ring, H i is an indecomposable endoregular R -modulefor all 1 ≤ i ≤ m . By hypothesis, H j ∼ = H k for m i − < j, k ≤ m i where m i = P ik =0 ℓ k and ℓ = 0 and for each 1 ≤ i ≤ n . Hence M = M ⊕ · · · ⊕ M n where M i = L m i k = m i − +1 H k ∼ = H ( ℓ i ) i . Without loss of generality we can take a summand of each M i and assume that M = H ( ℓ )1 ⊕ · · · ⊕ H ( ℓ n ) n . Let ρ : H j → H i be any nonzero homomorphism for 1 ≤ i < j ≤ n .Assume that 0 = x ∈ H j such that ρ ( x ) = 0. Consider ρ ⊕ H ( ℓ j ) j → H ( ℓ i ) i . Then ( ρ ⊕ x ) = 0. This implies that ( ρ ⊕ ∈ l S ( x ) ∩ V ij = 0, a contradiction. Therefore x = 0.This implies that ρ is a monomorphism. Hence it is easy to see that H i is H j -Rickart forall 1 ≤ i, j ≤ n . (cid:3) Remark 4.7. If M is a finite direct sum of indecomposable endoregular modules, thenEnd R ( M ) is a semi-perfect ring. Corollary 4.8.
Suppose that M = L ni =1 H i with H i an indecomposable endoregular module, H i is H j -Rickart for all ≤ i, j ≤ n and H i ≇ H j for i = j . If there exists an ordering I n = { , , ..., n } for the class { H i } i ∈I n such that H i is H j -injective for all i < j , then M is a Rickart module and End R ( M ) is isomorphic to a upper triangular matrix ring D V V · · · V n D V · · · V n D · · · V n ... ... ... . . . ... · · · D n where D i is a division ring for ≤ i ≤ n , and V ij is a D i - D j -bimodule for all ≤ i < j ≤ n .In particular, End R ( M ) is a semiprimary PWD.Proof. It follows directly from [13, Corollary 2.13] and Theorem 4.6. (cid:3)
The next example illustrates Theorem 4.6 and Corollary 4.8.
Example 4.9.
Let F be a field and R = (cid:0) F F F (cid:1) . Consider the right R -module M = (cid:0) F F (cid:1) ⊕ (cid:0) F (cid:1) ⊕ (cid:0) F (cid:1) = (cid:0) F F (cid:1) ⊕ (cid:0) F (cid:1) (2) . Denote H = (cid:0) F F (cid:1) and H = (cid:0) F (cid:1) . Then End R ( H ) = F = End R ( H ), Hom R ( H , H ) = F and Hom R ( H , H ) = 0. Thus, M satisfies the condition (a) of Theorem 4.6. Hence theendomorphism ring of M is End R ( M ) ∼ = (cid:16) F F F F F F F (cid:17) . Moreover, H is H (2)2 -injective because H is simple, therefore M = H ⊕ H (2)2 is a Rickartmodule. In particular, R is a right Rickart ring.Inspired by the last example, we have the following proposition. Proposition 4.10.
Let M be a finite Σ -Rickart module and P be any simple module suchthat Hom R ( M, P ) = 0 . Then M ( ℓ ) ⊕ P ( n ) is a finite Σ -Rickart module for any ℓ, n > .Proof. Let k >
0. Hence ( M ( ℓ ) ⊕ P ( n ) ) ( k ) = M ( kℓ ) ⊕ P ( kn ) . It is clear that P ( kn ) is M ( kℓ ) -Rickart and by hypothesis, M ( kℓ ) is P ( kn ) -Rickart. Since P ( kn ) is semisimple, M ( kℓ ) is P ( kn ) -injective. It follows from [13, Corollary 2.13] that M ( kℓ ) ⊕ P ( kn ) is a Rickart module.Thus, M ( ℓ ) ⊕ P ( n ) is a finite Σ-Rickart module for any ℓ, n > (cid:3) Remark 4.11.
It follows from Proposition 4.10 that the module M in Example 4.9 is notjust Rickart, but finite Σ-Rickart. In particular, the ring R = (cid:0) F F F (cid:1) is right hereditary. Corollary 4.12.
The following conditions are equivalent for a ring R : (a) R = L ni =1 I ( ℓ i ) i with I i an endoregular right ideal and I i is I j -Rickart for all ≤ i, j ≤ n ; INITE Σ-RICKART MODULES (b) R is isomorphic to a formal matrix ring Mat ℓ ( D ) V · · · V n ℓ ( D ) · · · V n ... ... . . . ... · · · Mat ℓ n ( D n ) where D i is a division ring for ≤ i ≤ n , and for all ≤ i < j ≤ n , V ij is a Mat ℓ i ( D i ) - Mat ℓ j ( D j ) -bimodule satisfying l R ( x ) ∩ V ij = 0 for any = x ∈ I j . Inparticular, R is a semiprimary PWD. To illustrate the last results, we have the following examples:
Example 4.13. (i) Let K be a field. It follows from [1, Ch.I Lemma 1.12 and Ch. VIITheorem 1.7] that every path algebra K Q of a finite, connected and acyclic quiver Q satisfiesthe conditions of Corollary 4.12.(ii) Let K and F be division rings and U be any left K - right F -bimodule. Thenthe formal matrix ring R = (cid:0) K U F (cid:1) trivially satisfies the condition (b) of Corollary 4.12.Moreover, the decomposition R = (cid:0) K U (cid:1) ⊕ (cid:0) F (cid:1) makes R to be a hereditary semiprimaryPWD by Corollary 4.12 and Proposition 4.10.(iii) Let K be a field and consider the ring R = (cid:16) K K K K K (cid:17) . Then R has a decompositionin hollow endoregular right ideals R = (cid:16) K K K (cid:17) ⊕ (cid:16) K (cid:17) ⊕ (cid:16) K (cid:17) . Denote those summands by I , I and I from the left to the right respectively. Hence I and I are simple R -modules. By Corollary 4.12 and applying Proposition 4.10 twice,we have that R is a (semi-)hereditary semiprimary PWD. Moreover by Corollary 4.12(b), R ∼ = (cid:16) K K K K
00 0 K (cid:17) where the isomorphism is given by (cid:16) a b c d e (cid:17) (cid:16) c b d a
00 0 e (cid:17) .(iv) Let K be a field and K [ x ] be the polynomial ring with coefficients in K . Considerthe ring R = (cid:18) K h x i K K [ x ] / h x i K (cid:19) where h x i is the ideal generated by x . Hence R has adecomposition in indecomposable endoregular ideals R = (cid:16) K h x i
00 0 00 0 0 (cid:17) ⊕ (cid:16) K K [ x ] / h x i (cid:17) ⊕ (cid:16) K (cid:17) . Let I , I , I denote those summand from the left to the right respectively. Let 0 = f ( x ) ∈ K [ x ] / h x i then (cid:16) x
00 0 00 0 0 (cid:17) ∈ l R (cid:18) f ( x )0 0 0 (cid:19) ∩ I . Thus, R does not satisfies the condition (b) inCorollary 4.12. Note that R is a semiprimary ring. Definition 4.14 ([16, Definition 2.23]) . A family of modules { M α | α ∈ Λ } is said to be locally-semi-Transfinitely-nilpotent ( lsTn ) if for any subfamily M α i ( i ∈ N ) with distinct α i and any family of non-isomorphisms ϕ i : M α i → M α i +1 , and for every x ∈ M α , there exists n ∈ N (depending on x ) such that ϕ n · · · ϕ ϕ ( x ) = 0. Proposition 4.15.
Consider the following conditions for a Rickart module M : (i) M = L ni =1 H i with H i a hollow endoregular module; (ii) End R ( M ) is a semiprimary ring.Then (i) ⇒ (ii) . In addition, if M is finitely generated then the two conditions are equivalent.Proof. (i) ⇒ (ii) Since M is a Rickart module, H i is H j -Rickart. It follows from Theorem 4.6that S is semiprimary. In addition, suppose M is finitely generated. (ii) ⇒ (i) Suppose S = End R ( M ) is a semipri-mary ring. Then S is right perfect. From Lemma [22, 43.8] M has D condition. Also M is quasi-discrete because M is Rickart. Therefore M has an irredundant decomposition M = L ni =1 H i with H i a hollow module such that complements summands and is uniqueup to isomorphism by [16, Theorem 4.15]. Since M ( N ) = n M i =1 H ( N ) i and by [22, 43.8], Rad M ( N ) ≪ M ( N ) , the module M ( N ) is quasi-discrete and the family { H = H j | j ∈ N } ∪ · · · ∪ { H n = H n j | j ∈ N } satisfies lsTn by [16, Theorem 4.53 and Corollary 4.49]. Since H i is indecomposable Rickartfor all 1 ≤ i ≤ n , every nonzero endomorphism ϕ : H i → H i is a monomorphism. By lsTn, ϕ must be an isomorphism. Thus H i is an endoregular module for all 1 ≤ i ≤ n . (cid:3) With the following examples we will show that the hypothesis on M to be Rickart andon H i to be endoregular in Proposition 4.15 are not superfluous. Example 4.16. (i) Set M = Z as Z -module. It is clear that End Z ( M ) = Z is a semipri-mary ring and M is a hollow module. But M is neither Rickart nor endoregular.(ii) Let R = Z ( p ) be the localization of integers at a prime p and set M = R R . Then M is a finitely generated Rickart module. Note that M is a hollow R -module but is notendoregular. Moreover R = End R ( M ) is not a semiprimary ring because Rad( Z ( p ) ) = p Z ( p ) .It follows from an Auslander’s result [10, 5.72] that a semiprimary right semi-hereditaryring is right and left hereditary. The corollaries below give an extension of Auslander’sresult for the case of Σ-Rickart and fintie Σ-Rickart modules. Corollary 4.17.
Let M be a finite Σ -Rickart module. If M = L ni =1 H i with H i a hollowendoregular module then End R ( M ( ℓ ) ) ∼ = Mat ℓ ( S ) is a semiprimary ( right ) hereditary PWDfor all ℓ > where S = End R ( M ) .Proof. Let ℓ >
0. Then M ( ℓ ) = L ni =1 H ( ℓ ) i with each H i hollow endoregular. Since M is finite Σ-Rickart and finitely generated, M ( ℓ ) is a Rickart module. From Theorem 4.6End R ( M ( ℓ ) ) is a semiprimary PWD. On the other hand, End R ( M ( ℓ ) ) = Mat ℓ ( S ) is a semi-hereditary ring by Theorem 3.20. From [10, 5.72] End R ( M ( ℓ ) ) is a semiprimary hereditaryPWD. (cid:3) Corollary 4.18.
Let M be a finitely generated module such that M = L ni =1 H i with H i ahollow endoregular module. The following conditions are equivalent: (a) M is a Σ -Rickart module; (b) M is a finite Σ -Rickart module.Proof. (a) ⇒ (b) is clear. For, (b) ⇒ (a) End R ( M ) is a semiprimary hereditary ring, by Corol-lary 4.17. It follows from [11, Theorem 4.6] that M is a Σ-Rickart module. (cid:3) Corollary 4.19.
Consider the following conditions for a module M : (i) M is ( finite ) Σ -Rickart and M = L ni =1 H i with H i a hollow endoregular module; (ii) S = End R ( K ) is a semiprimary ( right ) hereditary ring for every K in add( M ) and S K is flat.Then (i) ⇒ (ii) . In addition, if M is finitely generated, then the two conditions are equivalent. INITE Σ-RICKART MODULES Proof. (i) ⇒ (ii) Let K ∈ add( M ). Then M ( n ) = K ⊕ L for some n >
0. Hence K is afinite Σ-Rickart module. On the other hand, M ( n ) has the cancellation property, by [16,Corollary 4.20]. Therefore K satisfies the hypothesis of Corollary 4.17. Thus End R ( K ) is asemiprimary (right) hereditary ring. (ii) ⇒ (i) follows from [11, Theorem 4.6] and Proposition4.15. (cid:3) AcknowledgmentsThe authors are very thankful to Research Institute of Mathematical Sciences, Chung-nam National University (CNU-RIMS), Republic of Korea, for the support of this re-search work. The first author gratefully acknowledges the support of this research workby the National Research Foundation of Korea (NRF) grant funded by the Korea govern-ment(MSIT)(2019R1F1A105988312)
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Gangyong Lee, Department of Mathematics Education, Chungnam National University
Yuseong-gu Daejeon 34134, Republic of Koreae-mail: lgy999 @ cnu.ac.krMauricio Medina-B´arcenas, Facultad de Ciencias F´ısico-Matem´aticas, Benem´eritaUniversidad Aut´onoma de Puebla, Av. San Claudio y 18 Sur, Col.San Manuel,Ciudad Universitaria, 72570, Puebla, M´exico.e-mail: mmedina @@