aa r X i v : . [ m a t h . A C ] F e b QUASI J -SUBMODULES ECE YETKIN CELIKEL AND HANI A. KHASHAN
Abstract.
Let R be a commutative ring with identity and M be a unitary R -module. The aim of this paper is to extend the notion of quasi J -idealsof commutative rings to quasi J -submodules of modules. We call a propersubmodule N of M a quasi J -submodule if whenever r ∈ R and m ∈ M suchthat rm ∈ N and r / ∈ ( J ( R ) M : M ), then m ∈ M - rad ( N ). We present variousproperties and characterizations of this concept (especially in finitely gener-ated faithful multiplication modules). Furthermore, we provide new classesof modules generalizing presimplifiable modules and justify their relation with(quasi) J -submodules. Finally, for a submodule N of M and an ideal I of R ,we characterize the quasi J -ideals of the idealization ring R (+) M . Introduction
All rings considered in this paper are commutative with identity elements, andall modules are unital. Let R be a ring and N be a submodule of an R -module M. By Z ( R ), reg ( R ) , N ( R ), J ( R ), Z ( M ) and M - rad ( N ) , we denote the set of zero-divisors of R , the set of regular elements in R , the nil radical of R , the Jacobianradical of R, the set of all zero divisors on M ; i.e. { r ∈ R : rm = 0 for some0 = m ∈ M } and the intersection of all prime submodules of M containing N ,respectively. For submodules N of M , the residual of N by M , ( N : M ) denotesthe ideal { r ∈ R : rM ⊆ N } . In particular, the ideal (0 : M ) is called theannihilator of M . Moreover, if I is an ideal of R , then the residual submodule N by I is [ N : M I ] = { m ∈ M : Im ⊆ N } . An R -module M is a multiplication moduleif every submodule N of M has the form IM for some ideal I of R . Equivalently, N = ( N : M ) M , [7].In 2015, Mohamadian [14] introduced the concept of r -ideals in commutativerings. A proper ideal I of a ring R is called an r -ideal if whenever a, b ∈ R , ab ∈ I and Ann ( a ) = 0 imply that b ∈ I where Ann ( a ) = { r ∈ R : ra = 0 } . As a subclassof r -ideals, Tekir et. al. [18] defined a proper ideal I of R to be n -ideal if whenever a, b ∈ R such that ab ∈ I and a / ∈ N ( R ), then b ∈ I . Later, Khashan and Bani-Ata[10] generalized this notion to J -ideals and J -submodules. A proper ideal I of R is said to be a J -ideal if whenever a, b ∈ R such that ab ∈ I and a / ∈ J ( R ), then b ∈ I . A proper submodule N of an R -module M is said to be a J -submodule ifwhenever r ∈ R and m ∈ M with rm ∈ N and r / ∈ ( J ( R ) M : M ), then m ∈ N .Generalizing the idea of J -ideals, as a very recent study [11], the class of quasi Date : January, 2021.2010
Mathematics Subject Classification.
Key words and phrases. quasi J -submodule, J -submodule, quasi J -ideal, quasi J -presimplifiable module, J -presimplifiable module.This paper is in final form and no version of it will be submitted for publication elsewhere. J -ideals has been defined and studied. A proper ideal I of R is said to be a quasi J -ideal if √ I = { x ∈ R : x n ∈ I for some n ∈ Z } is a J -ideal.The purpose of the present work is to generalize the notions of quasi J -idealsand J -submodules by defining and studying quasi J -submodules. We call a propersubmodule N of M a quasi J -submodule if whenever r ∈ R and m ∈ M suchthat rm ∈ N and r / ∈ ( J ( R ) M : M ), then m ∈ M - rad ( N ). In Section 2, weinvestigate many general properties of quasi J -submodules of an R -module M withvarious examples. J -submodules are obviously a quasi J -submodules, but the con-verse of this implication is not true in general (Example 1). Among many otherresults in this section, we investigate quasi J -submodules under various contexts ofconstructions such as homomorphic images, direct products and localizations (seePropositions 2, 3 and 4). In 1997, the notion of presimplifiable modules was firststudied by Anderson and and Valdes-Leon [4] as R -modules with property that Z ( M ) ⊆ J ( R ). Motivated from this concept, we introduce new generalizationsof presimplifiable modules which are quasi presimplifiable, J -presimplifiable andquasi J -presimplifiable modules. Example 3 is given to show that these general-izations are proper. The main result of this section (Theorem 1) gives a relationbetween quasi J -submodules (resp. J -submodules) and quasi presimplifiable (resp. J -presimplifiable) modules which enables us to construct more examples for quasi J -submodules. Precisely, a submodule N of an R -module M contained in J ( R ) M is a quasi J -submodule (resp. J -submodule) if and only if the quotient M/N is anon-zero quasi J -presimplifiable (resp. J -presimplifiable) R -module.The last section deals with quasi J -submodules in multiplication modules. InTheorem 2, Theorem 3 and Proposition 10, we present many properties and charac-terizations for quasi J -submodules of multiplication modules (especially, in finitelygenerated faithful multiplication modules). In particular, in such a module M , wecharacterize quasi J -submodules N as those in which the residual ideal ( N : M )is a quasi J -ideal. It is shown in Theorem 5 that every quasi J -submodule of an R -module M is contained in a maximal quasi J -submodule of M . Furthermore, if M is finitely generated faithful multiplication, then a maximal quasi J -submoduleof M is a J -submodule. For a submodule N of M and an ideal I of R , we finally(Theorem 6) give a characterization of J -ideals in the idealization ring R (+) M .2. General properties of Quasi J -submodules In this section, among other results concerning the general properties of quasi J -submodules, some characterizations of this notion will be investigated. Moreover,the relations among quasi J -submodules and some other types of submodules willbe clarified.First, we present the fundamental definition of quasi J -submodules which willbe studied in this paper. Definition 1.
Let R be a ring and let M be an R -module. A proper submodule N of M is called a quasi J -submodule if whenever r ∈ R and m ∈ M such that rm ∈ N and r / ∈ ( J ( R ) M : M ) , then m ∈ M - rad ( N ) . It is clear that any J -submodule of M is a quasi J -submodule. However, in thenext example we can see that the converse is not true in general. Example 1.
Consider the Z -module M = Z . Then one can directly see that M - rad ( h ¯0 i ) = h ¯2 i . Now, let r ∈ Z and ¯ k ∈ Z such that r · ¯ k ∈ h ¯0 i and r / ∈ ( J ( Z ) Z : UASI J -SUBMODULES 3 Z ) = (0 : Z ) = h i . Then clearly ¯ k ∈ h ¯2 i = M − rad ( h ¯0 i ) and so h ¯0 i is a quasi J -submodule. On the other hand, h ¯0 i is not a J -submodule since for example, · ¯2 = ¯0 but / ∈ (0 : Z ) and ¯2 = ¯0 . Following [12], a proper submodule N of an R -module M is called an r -submodule(resp. sr -submodule) if whenever am ∈ N with ann M ( a ) = 0 M (resp. ann R ( m ) =0 R ), then m ∈ N (resp. a ∈ ( N : M )) for each a ∈ R and m ∈ M .In general, the class of (quasi) J -submodules is not comparable with the classesof r -submodules, sr -submodules and prime submodules. Example 2. (1) The submodule N = h ¯0 i is a quasi J -submodule of the Z -module Z which is not an r -submodule, an sr -submodule or a prime submodule.(2) The submodule h i is a prime submodule of the Z -module Z which is not a(quasi) J -submodule.(3) The submodule N = h ¯2 i is an r -submodule and sr -submodule of the Z -module Z , [12, Example 1] which is not a (quasi) J -submodule. In the following result, we give a characterization for quasi J -submodules of an R -module M . Proposition 1.
Let M be an R -module and N be a proper submodule of M . Thefollowing are equivalent: (1) N is a quasi J -submodule of M . (2) If r ∈ R − ( J ( R ) M : M ) and K is a submodule of M with rK ⊆ N , then K ⊆ M - rad ( N ) . (3) If A * ( J ( R ) M : M ) is an ideal of R and K is a submodule of M with AK ⊆ N , then K ⊆ M - rad ( N ) .Proof. (1) ⇒ (2) Assume N is a quasi J -submodule. Suppose r ∈ R − ( J ( R ) M : M )and K is a submodule of M with rK ⊆ N . Then for all k ∈ K , rk ∈ N and so k ∈ M - rad ( N ). Therefore, K ⊆ M - rad ( N ) as needed.(2) ⇒ (3) Suppose AK ⊆ N for an ideal A * ( J ( R ) M : M ) and a submodule K of M . Then for r ∈ A − ( J ( R ) M : M ), we have rK ⊆ N and so by assumption, K ⊆ M - rad ( N ).(3) ⇒ (1) Let r ∈ R and m ∈ M such that rm ∈ N and r / ∈ ( J ( R ) M : M ). Then h r i h m i ⊆ N and h r i * ( J ( R ) M : M ) and so m ∈ h m i ⊆ M - rad ( N ) and we aredone. (cid:3) Lemma 1. [13]
Let ϕ : M −→ M be an R -module epimorphism.Then (1) If N is a submodule of M and ker ( ϕ ) ⊆ N , then ϕ ( M - rad ( N )) = M - rad ( ϕ ( N )) . (2) If K is a submodule of M , then ϕ − ( M - rad ( K )) = M - rad ( ϕ − ( K )) . Proposition 2.
Let ϕ : M −→ M be an R -module epimorphism. Then (1) If N is a quasi J-submodule of M with ker ( ϕ ) ⊆ N , then ϕ ( N ) is a quasiJ-submodule of M . (2) If K is a quasi J-submodule of M with ker ( ϕ ) ⊆ J ( R ) M , then ϕ − ( K ) is a quasi J-submodule of M .Proof. (1) Suppose ϕ ( N ) = M = ϕ ( M ) and let m ∈ M . Then ϕ ( m ) = ϕ ( n )for some n ∈ N and so m − n ∈ ker ( ϕ ) ⊆ N . So, m ∈ N and N = M which is a contradiction. Hence, ϕ ( N ) is proper in M . Let r ∈ R and m ∈ M such that rm ∈ ϕ ( N ) and r / ∈ ( J ( R ) M : M ). Choose m ∈ M such that ϕ ( m ) = m . Then rm = rϕ ( m ) = ϕ ( rm ) ∈ ϕ ( N ). Thus, ϕ ( rm − a ) = 0 forsome a ∈ N and so rm − a ∈ ker ( ϕ ) ⊆ N . It follows that rm ∈ N . Moreover, wehave r / ∈ ( J ( R ) M : M ). Indeed, if rM ⊆ J ( R ) M , then rM = rϕ ( M ) = ϕ ( rM ) ⊆ ϕ ( J ( R ) M ) = J ( R ) ϕ ( M ) = J ( R ) M which is a contradiction.Since N is a quasi J-submodule, then m ∈ M - rad ( N ). Thus, m = ϕ ( m ) ∈ ϕ ( M - rad ( N )) = M - rad ( ϕ ( N )) by Lemma 1 and ϕ ( N ) is a quasi J-submoduleof M .(2) Clearly, ϕ − ( K ) is proper in M . Let r ∈ R and m ∈ M such that rm ∈ ϕ − ( K ) and r / ∈ ( J ( R ) M : M ). Then rϕ ( m ) = ϕ ( rm ) ∈ K . Weprove that r / ∈ ( J ( R ) M : M ). Suppose on the contrary that rM ⊆ J ( R ) M .Then rϕ ( M ) ⊆ J ( R ) ϕ ( M ) and so ϕ ( rM ) ⊆ ϕ ( J ( R ) M ). Now, if x ∈ rM ,then ϕ ( x ) ∈ ϕ ( rM ) ⊆ ϕ ( J ( R ) M ) and hence x − t ∈ ker ( ϕ ) ⊆ J ( R ) M forsome t ∈ J ( R ) M . It follows that x ∈ J ( R ) M and rM ⊆ J ( R ) M which is acontradiction. Since K is a quasi J -submodule of M , then ϕ ( m ) ∈ M - rad ( K ).Therefore, m ∈ ϕ − ( M - rad ( K )) = M - rad ( ϕ − ( K )) by Lemma 1 and the resultfollows. (cid:3) Corollary 1.
Let N and L be submodules of an R -module M with L ⊆ N . If N is a quasi J -submodule of M , then N/L is a quasi J -submodule of M/L . Proposition 3.
Let M , M , ..., M k be R -modules and consider the R -module M = M × M × · · · × M k .(1) If N = N × N × · · · × N k is a quasi J -submodule of M , then N i is a quasi J -submodule of M i for all i such that N i = M i .(2) If N j is a quasi J -submodule of M j for some j ∈ { , , ..., k } , then N = M × M × · · · × N j × · · · × M k is a quasi J -submodule of M .Proof. (1) With no loss of generality, we assume N = M and prove that N isa quasi J -submodule of M . Let r ∈ R and m ∈ M such that rm ∈ N and r / ∈ ( J ( R ) M : M ). Then r. ( m, , ..., ∈ N and clearly r / ∈ ( J ( R ) M : M ). Itfollows that ( m, , ..., ∈ M - rad ( N ) and so m ∈ M - rad ( N ) as required.(2) With no loss of generality, suppose N is a quasi J -submodule of M . Let r ∈ R and ( m , m , ..., m k ) ∈ M × M × · · · × M k such that ( r.m , r.m , ..., r.m k ) = r. ( m , m , ..., m k ) ∈ N × M × · · · × M k and r / ∈ ( J ( R ) M : M ). Then rm ∈ N and clearly r / ∈ ( J ( R ) M : M ). Therefore, m ∈ M - rad ( N ) and then( m , m , ..., m k ) ∈ M - rad ( N × M × · · · × M k ). (cid:3) Remark 1. (1) If N and N are quasi J -submodules of R -modules M and M respectively, then N × N need not be a quasi J -submodule of M × M . For example ¯0 and are quasi J -submodules of the Z -modules Z and Z respectively. However, ¯0 × is not a quasi J -submodule of Z × Z as . (¯1 , ∈ ¯0 × but / ∈ (¯0 × Z × Z ) and (¯1 , / ∈ M - rad (¯0 ×
0) = 2 Z × .(2) The condition ker ( ϕ ) ⊆ N in (1) of Proposition 2 is not necessary. Indeed,let M and M be R -modules and ϕ : M × M → M be the projection epimorphism.If N and N are proper submodules of M and M and N × N is a quasi J -submodule of M × M , then ϕ ( N × N ) = N is a quasi J -submodule of M .However, ker ( ϕ ) = 0 × M * N × N . UASI J -SUBMODULES 5 Let I be a proper ideal of R and N be a submodule of an R -module M . In thefollowing proposition, the notations Z I ( R ) and Z N ( M ) denote the sets { r ∈ R : rs ∈ I for some s ∈ R \ I } and { r ∈ R : rm ∈ N for some m ∈ M \ N } . Proposition 4.
Let S be a multiplicatively closed subset of a ring R such that S − ( J ( R )) = J ( S − R ) and M be an R -module. Then (1) If N is a quasi J -submodule of M and S − N = S − M , then S − N is aquasi J -submodule of the S − R -module S − M . (2) If S − N is a quasi J -submodule of S − M and S ∩ Z ( J ( R ) M : M ) ( R ) = S ∩ Z M - rad ( N ) ( M ) = ∅ , then N is a quasi J -submodule of M .Proof. (1) Suppose that rs ms ∈ S − N . Then urm ∈ N for some u ∈ S. Since N is a quasi J -submodule, then either ur ∈ ( J ( R ) M : M ) or m ∈ M - rad ( N ). If ur ∈ ( J ( R ) M : M ), then rs = urus ∈ S − ( J ( R ) M : M ) = ( S − J ( R ) S − M : S − M ) = ( J ( S − R ) S − M : S − M ) . If m ∈ M - rad ( N ), then ms ∈ S − ( M - rad ( N )) = S − M - rad ( S − N ) and we are done.(2) Let r ∈ R , m ∈ M and rm ∈ N . Then r m ∈ S − N which implies that r ∈ ( J ( S − R ) S − M : S − M ) = S − ( J ( R ) M : M ) or m ∈ S − M - rad ( S − N ) = S − ( M - rad ( N )) . Hence, either ur ∈ ( J ( R ) M : M ) for some u ∈ S or vm ∈ M - rad ( N ) for some v ∈ S . Thus, our assumptions imply that either r ∈ ( J ( R ) M : M )or m ∈ M - rad ( N ) as needed. (cid:3) Following [9], a proper submodule N of an R -module M is called quasi primaryif whenever r ∈ R and m ∈ M such that rm ∈ N , then either r ∈ √ N : M or m ∈ M - rad ( N ). Proposition 5. If N is a quasi-primary submodule of an R -module M such that ( N : M ) ⊆ J ( R ) , then N is a quasi J -submodule of M. Proof.
Suppose N is quasi-primary and ( N : M ) ⊆ J ( R ). Let r ∈ R and m ∈ M such that rm ∈ N and r / ∈ ( J ( R ) M : M ) . Then r / ∈ J ( R ) and so by assumption r / ∈ √ N : M . It follows that m ∈ M - rad ( N ) as needed. (cid:3) Note that if ( N : M ) * J ( R ), then the above proposition need not be true. Forexample, consider the submodule N = h i of the Z -module Z . Then ( N : M ) = h i * J ( Z ). Moreover, N is primary (and so quasi-primary) which is clearly not aquasi J -submodule. In view of [11, Theorem 2], we have: Corollary 2. If N is a quasi-primary submodule of an R -module M such that ( N : M ) is a quasi J -ideal of R , then N is a quasi J -submodule of M. Following [16], a submodule N of an R -module M is called a pure submodule if rM ∩ N = rN for each r ∈ R . Moreover, N is called divisible if rN = N for each r ∈ Reg ( R ), the set of regular elements in R . Proposition 6.
Let N be a divisible J -submodule of an R -module M with ( J ( R ) M : M ) ⊆ Reg ( R ) . Then N is pure in M .Proof. It is clear that rN ⊆ rM ∩ N for each r ∈ R . Let r ∈ R and let n ∈ rM ∩ N .Then n = rm ∈ N for some m ∈ M . If r ∈ ( J ( R ) M : M ), then by assumption, rM ∩ N ⊆ N = rN . If r / ∈ ( J ( R ) M : M ), then m ∈ N since N is a J -submoduleof M and so n = rm ∈ rN . Thus, rM ∩ N = rN and N is pure in M . (cid:3) ECE YETKIN CELIKEL AND HANI A. KHASHAN
Synonymously to the Prime Avoidance Lemma for prime submodules, we have:
Proposition 7.
Let M be an R -module such that J ( R ) = ( J ( R ) M : M ) is a quasi J -ideal of R . Let N, N , N , ..., N k be submodules of M where N ⊆ k [ i =1 N i . Supposethat N j is a J -submodule (resp. quasi J -submodule) with ( N i : M ) * J ( R ) for all i = j . If N * k [ i = j N i , then N ⊆ N j (resp. N ⊆ M - rad ( N j ) ).Proof. Without loss of generality, assume that j = k . First, we show that N ∩ k − \ i =1 N i ! ⊆ N k . Let x ∈ N ∩ k − \ i =1 N i ! . Since N * k − [ i =1 N i , there exists anelement m ∈ N k but m / ∈ k − [ i =1 N i . Then clearly m + x ∈ N \ k − [ i =1 N i ! . Hence, m + x ∈ N k and so x ∈ N k . Now, since ( N i : M ) * J ( R ) for all i = k , thereis an element r i ∈ ( N i : M ) \ J ( R ) for all i = k . Put r = k − Y i =1 r i . Since J ( R ) is aprime ideal of R , [11, Corollary 2], then r / ∈ J ( R ). Put I = k − \ i =1 ( N i : M ). Then I * J ( R ) = ( J ( R ) M : M ) and IN ⊆ N ∩ k − \ i =1 N i ! ⊆ N k . Since N k is a (quasi) J -submodule, we conclude that N ⊆ N j (resp. N ⊆ M - rad ( N j )) by Proposition1. (cid:3) For an R -module M , consider the set of all zero divisors on M , Z ( M ) = { r ∈ R : rm = 0 for some 0 = m ∈ M } . Following [4], we call an R -module M pres-implifiable if whenever r ∈ R and m ∈ M such that rm = m , then m = 0 or r ∈ U ( R ). Equivalently, M is presimplifiable if and only if Z ( M ) ⊆ J ( R ). Werecall that the prime radical of an R -module M is the intersection of all prime sub-modules in M and is denoted by N il ( M ). It is known that for a submodule N of M ,there is a one to one correspondence between the prime submodules of M/N andthose of M containing N . Hence, we get m ∈ M - rad ( N ) ⇐⇒ m + N ∈ N il ( M/N ).More generally, let
N Z ( M ) = { r ∈ R : rm = 0 for some m / ∈ N il ( M ) } . Next,we define some generalizations of presimplifiable modules. Definition 2.
Let M be an R -module. (1) M is called quasi presimplifiable if N Z ( M ) ⊆ J ( R ) . (2) M is called J -presimplifiable if Z ( M ) ⊆ ( J ( R ) M : M ) . (3) M is called a quasi J -presimplifiable if N Z ( M ) ⊆ ( J ( R ) M : M ) . Example 3. (1)
Consider the Z -module M = Z p for a prime integer p . Then Z ( M ) = h p i = (0 : M ) = ( J ( Z ) M : M ) . Thus, M is a J -presimplifiablemodule that is not presimplifiable. (2) The Z -module M = Z is a quasi J -presimplifiable that is not J -presimplifiable.Indeed, we have Z ( M ) = 2 Z and N Z ( M ) = 4 Z = ( J ( Z ) M : M ) . UASI J -SUBMODULES 7 (3) Consider the Z (+) Z -module M = Z (+) Z . Then M is a quasi presimpli-fiable module that is not presimplifiable, see [11, Example 5] . In the following theorem, we characterize J -submodules (resp. quasi J -submodules)in terms of J -presimplifiable (resp. quasi J -presimplifiable) modules. Theorem 1.
Let N be a submodule of an R-module M with N ⊆ J ( R ) M . Then N is a quasi J -submodule (resp. J -submodule) if and only if M/N is a non-zeroquasi J -presimplifiable (resp. J -presimplifiable) R -module.Proof. Suppose N is a quasi J -submodule. Let r ∈ N Z ( M/N )) and choose m + N / ∈ N il ( M/N ) such that r ( m + N ) = N . Then rm ∈ N and m / ∈ M - rad ( N ) sinceotherwise, if m ∈ M - rad ( N ), then m + N ∈ N il ( M/N ), a contradiction. Since N is a quasi J -submodule, then r ∈ ( J ( R ) M : M ) = (( J ( R ) M ) /N : M/N ) =( J ( R )( M/N ) :
M/N ) as needed. Conversely, suppose
M/N is a non-zero quasi J -presimplifiable and let r ∈ R and m ∈ M such that rm ∈ N and r / ∈ ( J ( R ) M : M ) = ( J ( R )( M/N ) :
M/N ). Then r. ( m + N ) = N and r / ∈ N Z ( M/N ). Therefore,we must have m + N ∈ N il ( M/N ) and then m ∈ M - rad ( N ). The proof of the J -submodule case is similar. (cid:3) Corollary 3.
Let N be a submodule of an R -module M such that N ⊆ J ( R ) M and ( J ( R ) M : M ) = J ( R ) . Then the following statements are equivalent: (1) N is a (quasi) J -submodule. (2) M/N is a non-zero (quasi) J -presimplifiable. (3) M/N is a non-zero (quasi) presimplifiable.
Recall that for an R -module M , T ( M ) = { m ∈ M : ann R ( m ) = 0 } . It is clearthat if N is an sr -submodule of M , then N ⊆ T ( M ). Proposition 8.
Let M be an R -module.(1) If M is J -presimplifiable and N is an r -submodule of M , then N is a (quasi) J -submodule of M .(2) If N is an sr -submodule of M with T ( M ) = N ⊆ J ( R ) M , then N is a(quasi) J -submodule of M .Proof. (1) Suppose N is an r -submodule and let r ∈ R and m ∈ M such that rm ∈ N and r / ∈ ( J ( R ) M : M ). Since M is J -presimplifiable, then r / ∈ Z ( M ) andso clearly ann M ( r ) = 0. Hence, m ∈ N ⊆ M - rad ( N ) as N is an r -submodule andwe are done.(2) Suppose N is an sr -submodule of M with N = T ( M ). Let r ∈ R and m ∈ M such that rm ∈ N . If m / ∈ M - rad ( N ), then m / ∈ T ( M ) and so ann R ( m ) = 0. Byassumption, we get r ∈ ( N : M ) ⊆ ( J ( R ) M : M ) and N is a (quasi) J -submoduleof M . (cid:3) Quasi J -submodule in multiplication modules In this section we study quasi J -submodules in some special types of modules.We give several properties and characterizations of quasi J -submodules in finitelygenerated faithful multiplication modules. Moreover, we determine conditions ona submodule N of M and an ideal I of R for which I (+) N is a quasi J -ideal in R (+) M .We start by the following lemma. ECE YETKIN CELIKEL AND HANI A. KHASHAN
Lemma 2. [17]
Let M be a finitely generated faithful multiplication R -module, N be a proper submodule of M and I be an ideal of R . Then (1) M - rad ( N ) = √ N : M M . (2) IM : M = I . (3) ( IN : M ) = I ( N : M ) . In view of the properties in Lemma 2, we give the following characterizations ofquasi J -submodules of finitely generated faithful multiplication modules. Theorem 2.
Let I be an ideal of a ring R and N be a submodule of a finitelygenerated faithful multiplication R -module M . Then (1) I is a quasi J -ideal of R if and only if IM is a quasi J -submodule of M .(2) N is a quasi J -submodule of M if and only if ( N : M ) is a quasi J -ideal of R .(3) N is a quasi J -submodule of M if and only if N = IM for some quasi J -ideal of R .(4) If I is a quasi J -ideal of R and N is a quasi J -submodule of M , then IN is a quasi J -submodule of M . Proof. (1) Suppose I is a quasi J -ideal of R . If IM = M , then I = ( IM : M ) = R ,a contradiction. Thus, IM is proper in M . Now, let r ∈ R and m ∈ M such that rm ∈ IM and r / ∈ ( J ( R ) M : M ) = J ( R ). Then r (( m ) : M ) = (( rm ) : M ) ⊆ ( IM : M ) ⊆ ( √ IM : M ) = √ I . As I is a quasi J -ideal of R , we conclude that(( m ) : M ) ⊆ √ I . Thus, m ∈ (( m ) : M ) M ⊆ √ IM = M - rad ( IM ). Conversely,suppose IM is a quasi J -submodule of M . Then clearly I is proper in R . Let a, b ∈ R such that ab ∈ I and a / ∈ J ( R ) = ( J ( R ) M : M ). Since abM ∈ IM and IM is a quasi J -submodule, then bM ⊆ M - rad ( IM ) = √ IM . Therefore, b ∈ √ IM : M = √ I and I is a quasi J -ideal of R .(2) Follows by (1) since N = ( N : M ) M .(3) Follows by choosing I = ( N : M ) and using (2).(4) Suppose I is a quasi J -ideal of R and N is a quasi J -submodule of M . Now,( N : M ) is a quasi J -ideal of R by (2) and so I ( N : M ) is also quasi J -ideal by[11, Proposition 4]. Moreover, IN = I ( N : M ) M is proper in M since otherwise, I ( N : M ) = R , a contradiction. By using (1), we conclude that IN is a quasi J -submodule of M . (cid:3) However, the equivalence in (2) of Theorem 2 can not be achieved if M is notfinitely generated faithful multiplication. For example, consider the Z -module M = Z × Z and the submodule N = 2 Z × M . Then clearly ( N : M ) = 0 is a quasi J -ideal of Z , but N is not a quasi J -submodule of M . In fact, 2 . (1 , ∈ N butneither 2 ∈ ( J ( Z ) M : M ) = 0 nor (1 , ∈ M - rad ( N ) = N . Proposition 9.
Let N be a submodule of a faithful multiplication R -module M .Let I be a finitely generated faithful multiplication ideal of R . Then (1) If IN is a J -submodule of M , then either I is a J -ideal of R or N is a J -submodule of M . (2) If √ I is a finitely generated multiplication ideal of R and √ IN is a quasi J -submodule of M , then either I is a quasi J -ideal of R or N is a quasi J -submodule of M . UASI J -SUBMODULES 9 Proof. (1) If N = M , then ( IN : M ) = I ( N : M ) = IR = I is a J -ideal of R by [10,Corollary 3.4]. Suppose N M . Since I is finitely generated faithful multiplication,we have N = ( IN : M I ), [1, Lemma 2.4]. Hence, one can easily verify that( N : M ) = (( IN : M I ) : M ) = ( I ( N : M ) : I ). Let a, b ∈ R such that ab ∈ ( N : M )and a / ∈ J ( R ). Then Iab ⊆ I ( N : M ) = ( IN : M ) and so Ib ⊆ I ( N : M ) as( IN : M ) is a J -ideal. It follows that b ∈ ( I ( N : M ) : I ) = ( N : M ) and so( N : M ) is a J -ideal of R . The result follows again by [10, Corollary 3.4].(2) If N = M , then √ I = √ I ( N : M ) = ( √ IN : M ) is a quasi J -ideal of R by (2)of Theorem 2. It follows clearly that I is a quasi J -ideal. Suppose N M and noteagain by [1, Lemma 2.4] that M - rad ( N ) = ( √ I ( M - rad ( N )) : M √ I ). Let rm ∈ N and r / ∈ J ( R ) for r ∈ R and m ∈ M . Then √ Irm ⊆ √ IN and so √ Im ⊆ M - rad ( √ IN ) = √ I ( M - rad ( N )). It follows that m ∈ ( √ I ( M - rad ( N )) : M √ I ) = M - rad ( N ) and N is a quasi J -submodule of M . (cid:3) Theorem 3.
Let N be a proper submodule of a finitely generated faithful multipli-cation R -module M . The following are equivalent: (1) N is a quasi J -submodule.(2) M - rad ( N ) is a quasi J -submodule.(3) M - rad ( N ) is a J -submodule.(4) ( M - rad ( N ) : M h r i ) = M - rad ( N ) for all r / ∈ J ( R ). Proof. (1) ⇒ (2) Suppose N is a quasi J -submodule and let r ∈ R and m ∈ M suchthat rm ∈ M - rad ( N ) and r / ∈ ( J ( R ) M : M ) = J ( R ). Then rm ∈ √ N : M M and so r (( m ) : M ) = (( rm ) : M ) ⊆ ( √ N : M M : M ) = √ N : M . Since ( N : M )is a quasi J -ideal by Theorem 2, then (( m ) : M ) ⊆ √ N : M . It follows that m ∈ (( m ) : M ) M ⊆ √ N : M M = M - rad ( N ).(2) ⇒ (3) It is straightforward as M - rad ( M - rad ( N )) = M - rad ( N ) . (3) ⇒ (4) Let m ∈ ( M - rad ( N ) : M h r i ). Then rm ∈ M - rad ( N ) with r / ∈ ( J ( R ) M : M ) and so m ∈ M - rad ( N ) by our assumption (2). The other inclusion is clear.(4) ⇒ (1) Suppose that rm ∈ N and r / ∈ ( J ( R ) M : M ) . Then rm ∈ M - rad ( N )and r / ∈ J ( R ) which imply that m ∈ ( M - rad ( N ) : M < r > ) = M - rad ( N ) . Thus, N is a quasi J -submodule. (cid:3) In general the equivalences in Theorem 3 need not be true if M is not finitelygenerated faithful multiplication. For example, while h ¯0 i is a quasi J-submodule inthe Z -module M = Z , M - rad ( h ¯0 i ) = h ¯2 i is not a J -submodule since for example2 · ¯1 ∈ h ¯2 i while 2 / ∈ (¯0 : Z ) and ¯1 / ∈ h ¯2 i .In view of Theorem 2 and Theorem 3, we also have: Proposition 10.
Let M be a finitely generated faithful multiplication R -module.For any submodule N of M , the following statements are equivalent. (1) N is a quasi J -submodule of M ..(2) p ( N : M ) is a J -ideal of R .(3) p ( N : M ) is a quasi J -ideal of R .(4) ( N : M ) is a quasi J -ideal of R . Proposition 11.
Let N , K and L be submodules of an R -module M and I be anideal of R with I * ( J ( R ) M : M ) . Then (1) If K and L are quasi J -submodules of M with IK = IL , then M - rad ( K ) = M - rad ( L ) . (2) If IN is a quasi J -submodule of a finitely generated faithful multiplicationmodule M , then N is a quasi J -submodule of M . Proof. (1) Suppose that IK = IL . Then IK ⊆ L and I * ( J ( R ) M : M ) implythat K ⊆ M - rad ( L ) by Proposition 1 and M - rad ( K ) ⊆ M - rad ( M - rad ( L )) = M - rad ( L ). Similarly, we conclude that M - rad ( L ) ⊆ M - rad ( K ), so the equality holds.(2) Let IN be a quasi J -submodule of M . Since IN ⊆ IN and I * ( J ( R ) M : M ),we conclude that N ⊆ M - rad ( IN ) . Hence, M - rad ( N ) = M - rad ( IN ) which isclearly a J -submodule of M by Theorem 3. It follows again by Theorem 3, that N is a quasi J -submodule of M . (cid:3) Lemma 3. [1]
Let I be a faithful multiplication ideal of a ring R and M be a faithfulmultiplication R -module. Then (1) For every submodule N of IM , we have ( IM ) - rad ( N ) = I ( M - rad ( N : M I )) . (2) If N is a submodule of M and I is finitely generated, then N = ( IN : M I ) . Proposition 12.
Let I be a faithful multiplication ideal of a ring R and M be afaithful multiplication R -module. If N is a quasi J -submodule of IM , then ( N : M I ) is a quasi J -submodule of M . Moreover, the converse is true if R is quasi-local.Proof. Suppose N is a quasi J -submodule of IM . Then N IM and so clearly,( N : M I ) M . Let r ∈ R and m ∈ M such that rm ∈ ( N : M I ) and r / ∈ ( J ( R ) M : M ).Then rmI ⊆ N and by Lemma 3 r / ∈ ( J ( R ) IM : IM ), hence mI ⊆ ( IM )- rad ( N ) = I ( M - rad ( N : M I )). It follows by Lemma 3 that m ∈ ( I ( M - rad ( N : M I )) : I ) = M - rad ( N : M I ). Therefore, ( N : M I ) is a quasi J -submodule of M . Now, suppose R is quasi-local and ( N : M I ) is a quasi J -submodule of M . Then clearly, N isproper in M and I = h a i is principal, see [3]. Let r ∈ R and m ∈ IM such that rm ∈ N and r / ∈ ( J ( R ) IM : IM ). Since I = h a i , then we may write m = am for some m ∈ M . Hence, rm ∈ ( N : M I ) and clearly r / ∈ ( J ( R ) M : M ). So, m ∈ M - rad (( N : M I )) as ( N : M I ) is a quasi J -submodule of M . Again byLemma 3, we have m = am ∈ I ( M - rad ( N : M I )) = ( IM )- rad ( N ) and the resultfollows. (cid:3) A submodule N of an R -module M is said to be small (or superfluous) in M ,abbreviated N ≪ M , in case for any submodule K of M , N + K = M implies K = M. Proposition 13.
Every quasi J -submodules of a finitely generated faithful multi-plication R -module is small.Proof. Let N be a quasi J -submodule of an R -module M and K be a submoduleof M with N + K = M . Then clearly ( N : M ) + ( K : M ) = ( N + K : M ) = R and ( N : M ) is a quasi J -ideal of R by Theorem 2. Hence ( K : M ) = R by [11,Proposition 4] and so K = M as desired. (cid:3) Let M be an R -module and N be a submodule of M . We denote the intersectionof all maximal submodules of M by J ( M ). In particular, by J ( N ), we denote theintersection of all maximal submodules of M containing N . It is well known thatIf M is finitely generated faithful multiplication, then J ( M ) = J ( R ) M , [8]. Inparticular, we have J ( N ) = J ( N : M ) M . UASI J -SUBMODULES 11 In the next two theorems, we obtain more characterizations for quasi J -submodulesin finitely generated faithful multiplication modules. Theorem 4.
Let N be a J -submodule of a finitely generated faithful multiplication R -module M. Then the following statements are equivalent: (1) N is a quasi J -submodule of M. (2) N ⊆ J ( M ) and if whenever r ∈ R and m ∈ M with rm ∈ N and r / ∈ ( J ( N ) : M ), then m ∈ M - rad ( N ). Proof. (1) ⇒ (2) Suppose N is a quasi J -submodule. since ( N : M ) is a quasi J -ideal by Theorem 2, then ( N : M ) ⊆ J ( R ), [11, Theorem 2]. Thus, N = ( N : M ) M ⊆ J ( R ) M = J ( M ). Moreover, let r ∈ R and m ∈ M with rm ∈ N and r / ∈ ( J ( N ) : M ). Then r / ∈ ( J ( M ) : M ) = ( J ( R ) M : M ) as clearly J ( M ) ⊆ J ( N )and so m ∈ M - rad ( N ) by assumption.(2) ⇒ (1) If N = M , then J ( M ) = M , a contradiction. Let r ∈ R and m ∈ M with rm ∈ N and r / ∈ ( J ( R ) M : M ) = J ( R ). Since N ⊆ J ( M ), then one can easilysee that J ( N ) ⊆ J ( J ( M )) = J ( M ) and so J ( M ) = J ( N ). Thus, r / ∈ ( J ( N ) : M )and so m ∈ M - rad ( N ) as required. (cid:3) Recall that If M is a multiplication R -module and N = IM , K = JM aretwo submodules of M , then the product N K of N and K is defined as N K =( IM )( JM ) = ( IJ ) M . In particular, if m , m ∈ M , then m m = h m i h m i . Proposition 14.
Let M be a finitely generated faithful multiplication R -moduleand N a proper submodule of M . Then N is a quasi J -submodule of M if andonly if whenever K and L are submodules of M with KL ⊆ N , then K ⊆ J ( M ) or L ⊆ M - rad ( N ) . Proof.
Suppose K = IM and L = JM for some ideals I and J of R and KL ⊆ N .Then I ( JM ) ⊆ N and so I ⊆ ( J ( R ) M : M ) = J ( R ) or L = JM ⊆ M - rad ( N )by Proposition 1. Thus, K ⊆ J ( R ) M = J ( M ) or L ⊆ M - rad ( N ). Conversely, let A * ( J ( R ) M : M ) = J ( R ) be an ideal of R and L be a submodule of M with AL ⊆ N. Then the result follows by putting K = AM and using again Proposition1. (cid:3) Corollary 4.
Let N be a proper submodule of a finitely generated faithful multipli-cation R -module M . Then N is a quasi J -submodule of M if and only if whenever m , m ∈ M such that m m ∈ N , then m ∈ J ( M ) or m ∈ M - rad ( N ) . Proposition 15.
Let M be a finitely generated faithful multiplication R -moduleand let S be a subset of R with S * J ( R ) . If N is a quasi J -submodule of M , then ( N : M S ) is a quasi J -submodule of M. Proof.
First, we prove that ( N : M S ) is proper in M . Suppose ( N : M S ) = M andlet m ∈ M . Then Sm ⊆ N and since N is a quasi J -submodule, we get m ∈ M - rad ( N ). Thus, M = M - rad ( N ) = N , a contradiction. Now, similar to the proof of(4) in Theorem 3, one can prove that ( M − rad ( N ) : M S ) = M − rad ( N ). Supposethat rm ∈ ( N : M S ) and r / ∈ J ( R ) . Then rSm ⊆ N and so Sm ⊆ M − rad ( N ) as N is a quasi J -submodule. It follows that m ∈ ( M − rad ( N ) : M S ) = M − rad ( N ) ⊆ M − rad ( N : M S ) as required. (cid:3) A proper submodule N of an R -module M is called a maximal quasi J -submoduleif there is no quasi J -submodule which contains N properly. Theorem 5.
Every quasi J -submodule of an R -module M is contained in a max-imal quasi J -submodule of M . Moreover, if M is finitely generated faithful multi-plication, then a maximal quasi J -submodule of M is a J -submodule.Proof. Suppose that N is a quasi J -submodule of M and Set Ω = { N α : N α is aquasi J -submodule of M , α ∈ Λ } . Then Ω = ∅ . Let N ⊆ N ⊆ · · · be any chainin Ω. We show that ∞ [ i =1 N i is a quasi J -submodule of M . Suppose rm ∈ ∞ [ i =1 N i for r ∈ R , m ∈ M and r / ∈ ( J ( R ) M : M ). Then rm ∈ N j for some j ∈ N which impliesthat m ∈ M - rad ( N j ) ⊆ M − rad ( ∞ [ i =1 N i ). Since also ∞ [ i =1 N i is clearly proper, then ∞ [ i =1 N i is a quasi J -submodule which is an upper bound of the chain { N i : i ∈ N } .By Zorn’s Lemma, Ω has a maximal element which is a maximal quasi J -submoduleof M . Now, let K be a maximal quasi J -submodule of M . Suppose that rm ∈ K and r / ∈ J ( R ). Then ( K : M r ) is also a quasi J -submodule of M by Proposition 15.Thus, the maximality of K implies that m ∈ ( K : M r ) = K and we are done. (cid:3) In view of Theorem 5, we have the following.
Corollary 5.
Let M be a finitely generated faithful multiplication R -module. Thenthe following statements are equivalent: (1) J ( M ) is a J -submodule of M . (2) J ( M ) is a quasi J -submodule of M . (3) J ( M ) is a prime submodule of M .Proof. (1) ⇒ (2) is clear.(2) ⇒ (1) It follows since J ( M ) is the unique maximal quasi J -submodule of M by Theorem 4.(2) ⇔ (3) Since M − rad ( J ( M )) = J ( M ), the claim is clear. (cid:3) In the following proposition, we prove that J -submodule property passes to afinite intersection and product. Proposition 16.
Let M be a multiplication R -module and N , N , ..., N k be quasi J -submodules of M . Then so are k \ i =1 N i and k Q i =1 N i .Proof. Suppose that rm ∈ k \ i =1 N i and r / ∈ ( J ( R ) M : M ). Then rm ∈ N i for all i =1 , ..., k which gives m ∈ M - rad ( N i ) for all i = 1 , ..., k. Since k \ i =1 M - rad ( N i ) = M - rad k \ i =1 N i ! [2, Theorem 15 (3)], we conclude that k \ i =1 N i is a quasi J -submodule.By using the similar argument and the equality k Q i =1 M - rad ( N i ) = M - rad k \ i =1 N i ! [15, Proposition 2.14.], k Q i =1 N i is also a quasi J -submodules of M . (cid:3) UASI J -SUBMODULES 13 The converse of the above proposition can be achieved under certain conditions.
Proposition 17.
Let M be a finitely generated faithful multiplication R -moduleand N , N , ..., N k be quasi primary submodules of M such that √ N i : M are notcomparable for all i = 1 , ..., k. If k \ i =1 N i or k Q i =1 N i is a quasi J -submodule of M , then N i is a quasi J -submodule of M for each i = 1 , ..., k .Proof. Suppose that N i ( i = 1 , ..., k ) is a quasi primary submodule of M. Then( N i : M ) is a quasi primary ideal of R for all i = 1 , ..., k , [9, Lemma 2.12]. If k \ i =1 N i is a quasi J -submodule, we conclude from Theorem 2 that k \ i =1 N i : M ! = k \ i =1 ( N i : M ) is a quasi J -ideal of R . Hence, ( N i : M ) is a quasi J -ideal of R forall i = 1 , ..., k by [11, Proposition 5]. Thus, N i is a quasi J -submodule of M forall i = 1 , ..., k by Theorem 2. The proof of the finite product case is similar since (cid:18) k Q i =1 N i : M (cid:19) = k Q i =1 ( N i : M ) and by using Theorem 2 and [11, Proposition 6]. (cid:3) Let R be a ring and M be an R -module. The idealization ring of M is theset R (+) M = R ⊕ M = { ( r, m ) : r ∈ R , m ∈ M } with coordinate-wise additionand multiplication defined as ( r , m )( r m ) = ( r r , r m + r m ). If I is anideal of R and N a submodule of M , then I (+) N is an ideal of R (+) M if andonly if IM ⊆ N . It is well known that if I (+) N is an ideal of R (+) M , then p I (+) N = √ I (+) M . Moreover, we have J ( R (+) M ) = J ( R )(+) M , [6]. Next, wecharacterize quasi J -ideals in any idealization ring R (+) M . Theorem 6.
Let I be an ideal of of a ring R and N be a submodule of an R -module M . Then I (+) N is a quasi J -ideal of R (+) M if and only if I is a quasi J -ideal of R .Proof. Suppose I (+) N is a quasi J -ideal of R (+) M and let a, b ∈ R such that ab ∈ I and a / ∈ J ( R ). Then ( a, b, ∈ I (+) N and ( a, / ∈ J ( R (+) M ). Therefore,( b, ∈ p I (+) N = √ I (+) M and so b ∈ √ I as needed. Conversely, suppose I is aquasi J -ideal of R . Let ( r , m ) , ( r , m ) ∈ R (+) M such that ( r , m ) ( r , m ) =( r r , r m + r m ) ∈ I (+) N and ( r , m ) / ∈ J ( R (+) M ) = J ( R )(+) M . Then r r ∈ I and r / ∈ J ( R ) which imply that r ∈ √ I . Thus, ( r , m ) ∈ √ I (+) M = p I (+) N and I (+) N is a quasi J -ideal of R (+) M . (cid:3) In view of Theorem 6, we have
Corollary 6.
Let I be an ideal of of a ring R and M be a finitely generated faithfulmultiplication R -module. If IM is a quasi J -submodule of M , then I (+) N is aquasi J -ideal of R (+) M for any submodule N of M .Proof. The result follows by Theorem 6 and (1) of Theorem 2. (cid:3)
We note that if I (+) N is a quasi J -ideal of R (+) M , then N need not be aquasi J -submodule of M . For example, while 0(+)0 is a quasi J -ideal of Z (+) Z by Theorem 6, but 0 is not quasi J -submodule of Z . For example, 2 . ¯3 = ¯0 but2 / ∈ ( J ( Z ) Z : Z ) = h i and ¯3 / ∈ M - rad (¯0) = ¯0. References [1] M. M. Ali, Residual submodules of multiplication modules, Beitr Algebra Geom, 46 (2)(2005), 405–422.[2] M. M. Ali, Idempotent and nilpotent submodules of multiplication modules, Comm. Algebra,36 (2008), 4620-4642.[3] D. D. Anderson, Multiplication ideals, multiplication rings and the ring R(x), Can. J. Math.,28 (4) (1976), 760-768.[4] D.D. Anderson and S. Valdes-Leon, Factorization in commutative rings with zero divisorsII, Factorization in integral domains, Lecture Notes in Pure and Appl. Math., 189, (1997),197-219.[5] D. D. Anderson, M. Axtell, S.J. Forman, J. Stickles, When are associates unit multiples?,Rocky Mountain J. Math., 34 (2004), 811-828.[6] D. D. Anderson, M. Winders, Idealization of a Module, Journal of Commutative Algebra, 1(1) (2009), 3-56.[7] A. Barnard, Multiplication modules. J. Algebra 71 (1981), 174–178.[8] Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. in Algebra, 16 (1988), 755-779.[9] F. M. Hosein, S. Mohdi, Quasi-primary Submodules Satisfying the Primeful Property I,Hacet. J. Math. Stat., 45 (5) (2016), 1421-1434.[10] H. A. Khashan, A. B. Bani-Ata, J -ideals of commutative rings, International ElectronicJournal of Algebra, 29 (2021), 148-164.[11] H. A. Khashan, E. Yetkin Celikel, Quasi J -ideals of commutative rings, (submitted).[12] S. Koc, U. Tekir, r -Submodules and sr -Submodules, Turk. J. Math. 42 (2018), 1863–1876.[13] C. P. Lu, M-radicals of submodules in modules, Math. Japonica, 34 (2) (1989), 211-219.[14] R. Mohamadian, r-ideals in commutative rings, Turkish Journal of Mathematics, 39 (2015),733-749.[15] H. Mostafanasab, E. Yetkin Celikel, U. Tekir, A. Y. Darani, On 2-absorbing primary sub-modules over commutative rings, An. St. Ovidius Constanta, Seria Matematica 24 (1) (2016),335-351.[16] P. Ribenboim, Algebraic Numbers. Wiley, 1972.[17] P. Smith, Some remarks on multiplication modules, Arch. Math., 50 (1988), 223-235.[18] U. Tekir, S. Koc, K. H. Oral, n -ideals of Commutative Rings, Filomat, 31 (10) (2017), 2933-2941. Department of Electrical-Electronics Engineering, Faculty of Engineering, HasanKalyoncu University, Gaziantep, Turkey.
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