An extension of S-artinian rings and modules to a hereditary torsion theory setting
aa r X i v : . [ m a t h . A C ] J a n An extension of S-artinian rings and modules to ahereditary torsion theory setting
P. Jara
Department of AlgebraUniversity of Granada
Abstract
For any commutative ring A we introduce a generalization of S –artinian ringsusing a hereditary torsion theory σ instead of a multiplicative closed subset S ⊆ A .It is proved that if A is a totally σ –artinian ring, then σ must be of finite type, and A is totally σ –noetherian. Introduction In [ ] , the authors study the problem of determining the structure of the polynomial ring D [ X ] over an integral domain D with field of fractions K , looking for the structure of theEuclidean domain K [ X ] . In particular, an ideal a ⊆ D [ X ] is said to be almost principal whenever there exist a polynomial F ∈ a , of positive degree, and an element 0 = s ∈ D such that a s ⊆ F D [ X ] ⊆ a . The integral domain D is an almost principal domain whenever every ideal a ⊆ D [ X ] , which extends properly to K [ X ] , is almost principal.Noetherian and integrally closed domains are examples of almost principal domains.Later, in [ ] , the authors extend this notion to non–necessarily integral domains in defin-ing, for a given multiplicatively closed subset S ⊆ A of a ring A , an ideal a ⊆ A to be S –finite if there exist a finitely generated ideal a ′ ⊆ a and an element s ∈ S such that a s ⊆ a ′ , and define a ring A to be S –noetherian whenever every ideal a ⊆ A is S –finite.Many authors have worked on S –noetherian rings and related notions, and shown rele-vant results on its structure. See for instance [
3, 7, 10, 11, 12, 14 ] . January 26, 2021; [email protected]
Mathematics Subject Classification:
Key words: noetherian ring and module, artinian ring and module. [ ] , the author study S –artinian rings, dualizing the former notion of S –noetherianring, and give some characterization of S –artinian rings in terms of finite cogenerationwith respect to S . Our aim is to show that this theory is part of a more general theoryinvolving hereditary torsion theories. In particular, we show that if A is totally σ –artinian,then the hereditary torsion theory σ is of finite type, and, in addition, A it is totally σ –noetherian.The background will be the hereditary torsion theories on a commutative (and unitary)ring A , see [
4, 13 ] , and Mod − A denotes the category of A –modules. Thus, a hereditarytorsion theory σ in Mod − A is given by one of the following objects:(1) a torsion class T σ , a class of modules which is closed under submodules, homomor-phic images, direct sums and group extensions,(2) a torsionfree class F σ , a class of modules which is closed under submodules, es-sential extensions, direct products and group extensions,(3) a Gabriel filter of ideals L ( σ ) , a non–empty filter of ideals satisfying that every b ⊆ A , for which there exists an ideal a ∈ L ( σ ) such that ( b : a ) ∈ L ( σ ) , for every a ∈ a , belongs to L ( σ ) .(4) a left exact kernel functor σ : Mod − A −→ Mod − A .The relationships between these notions are the following. If σ is the left exact kernelfunctor, then T σ = { M ∈ Mod − A | σ M = M } , F σ = { M ∈ Mod − A | σ M = } , L ( σ ) = { a ⊆ A | A / a ∈ T σ } .If L is the Gabriel filter of σ , and T the torsion class, for any A –module M we have: σ M = { m ∈ M | ( m ) ∈ L } = P { N ⊆ M | N ∈ T } . Example 0.1. (1) Let Σ ⊆ A be a multiplicatively closed subset, there exists a hereditarytorsion theory, σ Σ , defined by L ( σ Σ ) = { a ⊆ A | a ∩ Σ = ∅ } .Observe that σ Σ has a filter basis constituted by principal ideals. Every hereditarytorsion theory σ such that L ( σ ) has a filter basis of principal ideals is called a prin-cipal hereditary torsion theory. We can show that there is a correspondence betweenprincipal hereditary torsion theories in Mod − A , and saturated multiplicatively closedsubsets in A .(2) Let A be a set of finitely generated ideals of a ring A , then L = { b ⊆ A | there exists a , . . . , a t ∈ A such that a · · · a t ⊆ b } is a Gabriel filter. 2his paper is organized in sections. In the first one we introduce the main subject: to-tally σ –artinian rings and modules, and show examples, their first properties, and thedecisive fact: if A is a totally σ –artinian ring, then σ is a finite type hereditary torsiontheory. In section two we deal with scalar extensions, which will be useful for studyinglocal properties. In section three we give an extra characterization of totally σ –artinianrings and modules with the minimal conditions we found out. In the fourth section, westudy we study the behaviour of prime ideals in relation with totally σ –artinian modules.Sections five and six is devoted to establish the necessary background to show that everytotally σ –artinian rings is also totally σ –noetherian. σ –artinian rings and modules For any σ –torsion finitely generated A –module M , if M = m A + · · · + m t A , since ( m i ) ∈ L ( σ ) , for any i =
1, . . . , t , then h : = ∩ ti = ( m i ) ∈ L ( σ ) , and satisfies M h = σ –torsion non–finitely generated A –modules.Therefore, we shall define an A –module M to be totally σ –torsion whenever there exists h ∈ L ( σ ) such that M h =
0. The notion of totally torsion appears, for instance, in [ ] .For any ideal a ⊆ A we have two different notions of finitely generated ideals relative to σ :(1) a ⊆ A is σ –finitely generated whenever there exists a finitely generated ideal a ′ ⊆ a such that a / a ′ is σ –torsion.(2) a ⊆ A is totally σ –finitely generated whenever there exists a finitely generated ideal a ′ ⊆ a such that a / a ′ is totally σ –torsion.In the same way, for any ring A we have two different notions of noetherian ring relativeto σ :(1) A is σ –noetherian if every ideal is σ –finitely generated.(2) A is totally σ –noetherian whenever every ideal is totally σ –finitely generated. Example 1.1. (1) Every finitely generated ideal is totally σ –finitely generated and everytotally σ –finitely generated ideal is σ –finitely generated.(2) Let S ⊆ A be a multiplicatively closed subset, an ideal a ⊆ A is S –finite if, and onlyif, it is totally σ S –finitely generated. The ring A is S –noetherian if, and only if, A istotally σ S –noetherianWe may dualize this notions, thus, if A is a ring and σ a hereditary torsion theory in Mod − A ,(1) A is σ –artinian if every decreasing chain of ideals is σ –stable.32) A is totally σ –artinian if every decreasing chain of ideals is totally σ –stable.Being a decreasing chain of ideals a ⊇ a ⊇ · · · σ –stable whenever there exists an index m such that a s ⊆ σ a m , for every s ≥ m , i.e., every a s is σ –dense in a m , or equivalently, forevery x ∈ a m there exists h ∈ L ( σ ) such that x h ⊆ a s (observe that h depends of x and s ). Otherwise, the decreasing chain of ideals is totally σ –stable whenever there exist anindex m , and h ∈ L ( σ ) such that a m h ⊆ a s , for every s ≥ m . Lemma 1.2.
Forany ring A wehave: A isartinian ⇒ A istotally σ –artinian ⇒ A is σ –artinian.The notions of σ –artinian (resp. totally σ –artinian) and σ –noetherian (resp. totally σ -noetherian) ring can be extended to A –modules in an easy way.Trivial examples of totally σ –artinian modules are the totaly σ –torsion modules. Alsoevery artinian module is totally σ –artinian for every hereditary torsion theory σ .These two notions of torsion, and the derived notions from them, are completely differ-ent in its behaviour and its categorical properties. For instance, due to the definition,for any A –module M there exists a maximum submodule belonging to T σ , the submod-ule: σ M , and it satisfies M /σ M ∈ F σ . In the totally σ –torsion case we cannot assurethe existence of a maximal totally σ –torsion submodule. The existence of a maximum σ –torsion submodule allows us to build new concepts relative to σ as lattices, closureoperator and localization; concepts that we do not have in the totally σ –torsion case. Forinstance, the ring A is σ –artinian if, and only if, the lattice C ( A , σ ) = { a | A / a ∈ F σ } isan artinian lattice. Nevertheless, the totally σ –torsion case allows us to study arithmeticproperties of rings and modules which are hidden with that use of σ –torsion, and theseproperties are those which we are interested in studying.As we point out before, the σ –torsion allows us, for any A –module M , to define a lattice C ( M , σ ) = { N ⊆ M | M / N ∈ F σ } ,and in L ( M ) , the lattice of all submodules of M , a closure operator Cl M σ ( − ) : L ( M ) −→ C ( M , σ ) ⊆ L ( M ) , defined by the equation Cl M σ ( N ) / N = σ ( M / N ) . The elements in C ( M , σ ) are called the σ –closed submodules of M , and the lattice operations in C ( M , σ ) ,for any N , N ∈ C ( M , σ ) , are defined by N ∧ N = N ∩ N , N ∨ N = Cl M σ ( N + N ) .Dually, the submodules N ⊆ M such that M / N ∈ T σ are called σ –dense submodules.The set of all σ –dense submodules of M is represented by L ( M , σ ) , L ( σ ) in the case inwhich M = A . 4n the following, we assume A is a ring, Mod − A is the category of A –modules and σ is a hereditary torsion theory on Mod − A . Modules are represented by Latin letters: M , N , N , . . ., and ideals by Gothics letters: a , b , b , . . . Different hereditary torsion theorieswill be represented by Greek letters: σ , τ , σ , . . ., and induced hereditary torsion theoriesby adorned Greek letters: σ ′ , τ , . . .In order to establish equivalent condition to (totally) σ –artinian modules, we introducethe definition of finitely cogenerated A –module.(1) An A –module M is finitely cogenerated if for any family of submodules { N i | i ∈ I } such that ∩ i ∈ I N i =
0, there exists a finite subset J ∈ I such that ∩ j ∈ J N j = [ ] the author uses the notion of σ –finitely cogenerated modules;an A –module M is σ –finitely cogenerated if for any family of submodules { N i | i ∈ I } such that ∩ i ∈ I N i is σ –torsion there exists a finite subset J ⊆ I such that ∩ j ∈ J N j is σ –torsion.(3) In our case for totally σ –torsion, we define an A –module to be totally σ –finitelycogenerated whenever for every family of submodules { N i | i ∈ I } such that ∩ i ∈ I N i is totally σ –torsion there exists a finite subset J ⊆ I such that ∩ j ∈ J N j is totally σ –torsion, i.e., there exists h ∈ L ( σ ) such that ( ∩ j ∈ J N j ) h = Theorem 1.3.
Let A be a ring and σ a hereditary torsion theory in Mod − A , for any A –module M the followingstatementsare equivalent:(a) M istotally σ –artinian.(b) Everyquotient M / N of M istotally σ –finitelycogenerated.P ROOF (a) ⇒ (b). Let { N i / N | i ∈ I } be a family of submodules of M / N such that ∩ i ∈ I ( N i / N ) is totally σ –torsion. If H / N = ( ∩ i ∈ I N i ) / N = ∩ i ∈ I ( N i / N ) , then H / N is totally σ –torsion and ∩ i ∈ I ( N i / H ) =
0. We have a family { H i = N i / H | i ∈ I } of submodules of M / H such that ∩ i ∈ I H i =
0. By the hypothesis, M / H is σ –artinian, so there are maximalelements in the set Γ = {∩ j ∈ J H j | J ⊆ I is finite } .Let ∩ j ∈ J H j ∈ Γ be a minimal element in Γ . There exists h ∈ L ( σ ) such that for any i ∈ I \ J , we have ( ∩ j ∈ J H j ) h ⊆ ( ∩ j ∈ J H j ) ∩ N i ⊆ ∩ j ∈ J H j .In particular, ( ∩ j ∈ J H j ) h ⊆ ∩ i ∈ I H i =
0, and ∩ j ∈ J H j is totally σ –torsion.(b) ⇒ (a). Let N ⊇ N ⊇ · · · be a decreasing chain of submodules of M , and define N = ∩ n ∈ N N n . In M / N the family { N n / N | n ∈ N } satisfies ∩ n ∈ N ( N n / N ) =
0, hence thereexists J ⊆ N , finite, and h ∈ L ( σ ) such that ( ∩ j ∈ J ( N j / N )) h =
0, hence ( N k / N ) h = k = max ( J ) , and satisfies N k h ⊆ N . Therefore, the decreasing chain σ –stabilizes. ƒ σ –finitely cogenerated and σ –noetherian mod-ules are collected in the following result. Proposition 1.4. (1) Every submodule of a totally σ –finitely cogenerated A –modulealso is.(2) Foreverysubmodule N ⊆ M ,wehave: M istotally σ –artinianif,and onlyif, N and M / N are totally σ –artinian.(3) Finite directsums oftotally σ –artinian modulesalso are.Also we can build up examples of totally σ –artinian rings in considering hereditary tor-sion theories σ ≤ σ . Thus, we have the following lemma, whose proof is straightfor-ward. Lemma 1.5.
Let σ ≤ σ behereditarytorsiontheoriesin Mod − A ,and M an A –module.If M istotally σ –artinianthen M istotally σ –artinian.Regular elements have a particular behaviour with respect to totally σ –artinian rings. Lemma 1.6. If A is a totally σ –artinian ring, for any regular element a ∈ A , we have aA ∈ L ( σ ) .P ROOF If a ∈ A is regular, we consider the decreasing chain ( a ) ⊇ ( a ) ⊇ · · · . By thehypothesis, there exist an index m and h ∈ L ( σ ) such that ( a m ) h ⊆ ( a s ) for every s ≥ m .Thus, for every h ∈ h there exists x ∈ A such that a m h = a m + x , hence h = a x ∈ aA ,which means that h ⊆ aA , and aA ∈ L ( σ ) . ƒ As a consequence, the case of an integral domain is well understood. See [
12, Corol-lary 2.2 ] . Corollary 1.7.
Let A = D be an integral domain, if D is totally σ –artinian, then σ = σ D \{ } , theusual torsiontheoryon D .In particular, we have the following conclusions:(1) If p ⊆ D is a non–zero prime ideal of an integral domain D , and we consider thehereditary torsion theory σ D \ p , then D is never totally σ D \ p –artinian.(2) For every integral domain D , the hereditary torsion theory σ = σ D \{ } satisfies that D is σ –artinian, but non necessarily D is totally σ –artinian. Indeed, D is σ –artinianbecause D σ , the field of fractions of D , is artinian. Otherwise the following exampleshows that the converse non necessarily holds. Let D = Q [ X n | n ∈ N ] , and a n =( X · · · X n ) , for every n ∈ N ; the decreasing chain a ⊇ a ⊇ · · · satisfies that there isnot s ∈ D \ { } neither m ∈ N such that a m s ⊆ a s , for every s ≥ m .This example raises the following problem:6 roblem 1.8. Whichpropertiesarenecessarytoadd toa σ –artinianringtobeatotally σ –artinian ring?We refer to Theorem (4.5.) below. Corollary 1.9.
Let A be a totally σ –artinian ring, and Reg ( A ) be the set of all regularelementsof A ,then σ Reg ( A ) ≤ σ .P ROOF
It is a consequence of the well known fact that Reg ( A ) is a saturated multiplica-tively closed set. ƒ Let A be a ring and T = T ( A ) the total ring of fractions of A , i.e., the localization of A atReg ( A ) , the multiplicatively closed set of all regular elements, i.e., T = A σ Reg ( A ) . The aboveexample in (2) shows that non necessarily A must be totally σ Reg ( A ) –artinian, although itis σ Reg ( A ) –artinian.We said that an ideal a ⊆ A is invertible whenever a ( A : a ) = A , being ( A : a ) = { x ∈ T | a x ⊆ A } . Corollary 1.10. If A isatotally σ –artinianring,everyinvertibleideal a belongsto L ( σ ) P ROOF
Let a ⊆ A be an invertible ideal, we consider the decreasing chain a ⊇ a ⊇ · · · .By the hypothesis, there exist an index m and h ∈ L ( σ ) such that a m h ⊆ a s for every s ≥ m . In particular, a m h ⊆ a m + , hence h ⊆ a , and a ∈ L ( σ ) . ƒ Example 1.11.
Since invertible ideals are finitely generated ideals, they generate a here-ditary torsion theory, that we name σ inv , see (2) in Example (0.1.). If A is a totally σ –artinian ring, non necessarily A is totally σ inv –artinian.Indeed, we can consider the ring A = Q N Z , and the prime ideal p = Q n ≥ Z . Weknow that A is totally σ A \ p –artinian. Since A is a total ring of fractions, every non regularelement is invertible, hence T = T ( A ) = A , and the only invertible ideal is the proper A ,hence σ Reg ( A ) =
0. If A were totally σ Reg ( A ) –artinian then A must be exactly artinian, butobviously A is not artinian.In general, if A is a totally σ –artinian ring, we have one more property of the hereditarytorsion theory σ . Proposition 1.12. If A is a totally σ –artinian ring, the hereditary torsiontheory σ is offinite type.P ROOF
Since A is totally σ –artinian, it is σ –artinian and, by Hopkins’ Theorem, σ –noetherian, hence σ is of finite type. ƒ We are interested in proving stronger results: if A is totally σ –artinian, then A is totally σ –noetherian. 7 Scalar extensions
Let f : A −→ B be a ring map. For every hereditary torsion theory σ in Mod − A we maydefine a new hereditary torsion theory f ( σ ) in Mod − B being• L ( f ( σ )) = { b ⊆ B | f − ( b ) ∈ L ( σ ) } ,• T f ( σ ) = { M B | M A ∈ T σ } ,• F f ( σ ) = { M B | M A ∈ F σ } .In addition, sometimes, we shall impose the condition that every ideal of B is an extensionof an ideal of A , i.e., for every ideal b ⊆ B , there exists an ideal a ⊆ A such that b = f ( a ) B .With this condition, we have that the Gabriel filter L ( f ( σ )) can be described also as L ( f ( σ )) = { f ( a ) B | a ∈ L ( σ ) } . Lemma 2.1.
Let f : A −→ B be a ring map such that every ideal of B is an extension ofan ideal of A , and let σ be a hereditary torsion theory in Mod − A such that A is totally σ –artinian, then B istotally σ –artinian.P ROOF
Let b ⊇ b ⊇ · · · be a decreasing chain of ideals of B , and let a i ⊆ A be an idealsuch that b i = f ( a i ) B ; we can obtain a decreasing chain a ⊇ a ⊇ · · · of ideals A . By thehypothesis, there exist an index m and h ∈ L ( σ ) such that a m h ⊆ a s for every s ≥ m . Inconsequence, b m h B ⊆ b s for every s ≥ m , and B is totally f ( σ ) –artinian. ƒ Corollary 2.2.
Let A beatotally σ –artinian ring,thenwehave:(1) Ifforany ideal a ⊆ A we considerthecanonical projection p : A −→ A / a ,then A / a istotally p ( σ ) –artinian.(2) If for any multiplicatively closed subset Σ ⊆ A we consider the canonical map f : A −→ Σ − A , then Σ − A istotally f ( σ ) –artinian. Corollary 2.3.
Let a ⊆ A be an ideal, and p : A −→ A / a the canonical projection. Thefollowingstatementsare equivalent:(a) A istotally σ –artinian.(b) a is totally σ –artinian and A / a is totally σ –artinian (equivalently, it is totally p ( σ ) –artinian).And, as a consequence of Proposition(1.4.), we have: Corollary 2.4.
Let A beatotally σ –artinianring,theneveryfinitelygenerated A –moduleistotally σ –artinian. 8 The minimal condition
Let M be an A –module, after [ ] , we establish the following definitions:(1) Let N ⊆ L ( M ) be a family of submodules of M . An element N ∈ N is σ –minimal ifthere exists h ∈ L ( σ ) such that for every H ∈ N such that H ⊆ N we have N h ⊆ H .(2) The A –module M satisfies the σ -MIN condition if every nonempty family of sub-modules of M has σ -minimal elements.(3) A family N of submodules of M is σ –lower closed if for every submodule H ⊆ M such that there exist N ∈ N and h ∈ L ( σ ) satisfying N h ⊆ H , either equivalently N ⊆ ( H : h ) or equivalently ( H : N ) ∈ L ( σ ) , we have H ∈ N .We have the following characterization of totally σ –artinian modules. Proposition 3.1.
Let M bean A –module,the followingstatements areequivalent:(a) M istotally σ –artinian.(b) Everynonempty σ –lowerclosedfamilyofsubmodulesof M hasminimal elements.(c) Everynonemptyfamilyofsubmodulesof M has σ –minimal elements.If we have Σ ⊆ A a multiplicatively closed subset of A and σ = σ Σ , this proposition isTheorem 2.1 in [ ] .Let σ be a hereditary torsion theory in Mod − A ; an A –module M is(1) σ –finitely cogenerated if for any family of submodules { N i | i ∈ I } of M such that ∩ i ∈ I N i is σ –torsion, there exists a finite subset J ⊆ I such that ∩ j ∈ J N j is σ –torsion.(2) totally σ –finitely cogenerated if for any family of submodules { N i | i ∈ I } of M such that ∩ i ∈ I N i is totally σ –torsion, there exists a finite subset J ⊆ I such that ∩ j ∈ J N j is totally σ –torsion.(3) strongly totally σ –finitely cogenerated if for any family of submodules { N i | i ∈ I } of M such that ∩ i ∈ I N i =
0, there exists a finite subset J ⊆ I such that ∩ j ∈ J N j is totally σ –torsion.We are mainly interested in modules M such that every quotient M / N is σ –finitely cogen-erated (resp. totally σ –finitely cogenerated). For that reason we weaken the condition ∩ i ∈ I N i is σ –torsion (resp. totally σ –torsion) to simply consider that ∩ i ∈ I N i =
0, obtain-ing in this way the strongly totally σ –cogenerated modules. These modules will be alsouseful in order to compare hereditary torsion theories because if σ ≤ σ , an A –module M may be totally σ –finitely cogenerated and non necessarily totally σ –finitely cogen-erated.Now we can give another characterization of totally σ –artinian modules in the followingway. 9 heorem 3.2. Let A be a ring and σ a hereditary torsion theory in Mod − A , for any A –module M the followingstatementsare equivalent:(a) M istotally σ –artinian.(b) Everyquotient M / N of M isstronglytotally σ –finitely cogenerated.(c) Everyquotient M / N of M istotally σ –finitelycogenerated.P ROOF (a) ⇒ (b). Let { N i / N | i ∈ I } be a family of submodules of M / N such that ∩ i ∈ I ( N i / N ) =
0, hence ∩ i ∈ I N i = N . By the hypothesis, M / N is totally σ –artinian, sothere are maximal elements in the set Γ = {∩ j ∈ J N j | J ⊆ I is finite } .Let ∩ j ∈ J N j ∈ Γ be a minimal element in Γ . There exists h ∈ L ( σ ) such that for any i ∈ I \ J , we have ( ∩ j ∈ J N j ) h ⊆ ( ∩ j ∈ J N j ) ∩ N i ⊆ ∩ j ∈ J N j .In particular, ( ∩ j ∈ J N j ) h ⊆ ∩ i ∈ I N i =
0, and ∩ j ∈ J H j is totally σ –torsion.(a) ⇒ (c). Let { N i / N | i ∈ I } be a family of submodules of M / N such that ∩ i ∈ I ( N i / N ) is totally σ –torsion. If H / N = ( ∩ i ∈ I N i ) / N = ∩ i ∈ I ( N i / N ) , then H / N is totally σ –torsionand ∩ i ∈ I ( N i / H ) =
0. We have a family { H i = N i / H | i ∈ I } of submodules of M / H suchthat ∩ i ∈ I H i =
0. By the hypothesis, M / H is σ –artinian, so there are maximal elementsin the set Γ = {∩ j ∈ J H j | J ⊆ I is finite } .Let ∩ j ∈ J H j ∈ Γ be a minimal element in Γ . There exists h ∈ L ( σ ) such that for any i ∈ I \ J , we have ( ∩ j ∈ J H j ) h ⊆ ( ∩ j ∈ J H j ) ∩ H i ⊆ ∩ j ∈ J H j .In particular, ( ∩ j ∈ J H j ) h ⊆ ∩ i ∈ I H i =
0, and ∩ j ∈ J H j is totally σ –torsion. Therefore also ∩ j ∈ J ( N j / H ) and ∩ j ∈ J ( N j / N ) are totally σ –torsion(b) ⇒ (a), (c) ⇒ (a). Let N ⊇ N ⊇ · · · be a decreasing chain of submodules of M , anddefine N = ∩ n ∈ N N n . In M / N the family { N n / N | n ∈ N \ { }} satisfies ∩ n ∈ N ( N n / N ) = J ⊆ N , finite, and h ∈ L ( σ ) such that ( ∩ j ∈ J ( N j / N )) h =
0, hence ( N k / N ) h =
0, being k = max ( J ) , and satisfies N k h ⊆ N . Therefore the decreasing chain σ –stabilizes. ƒ Lemma 3.3.
Let M be an A –module and T ⊆ M be a totally σ –torsion submodule, thefollowingstatementsare equivalent:(a) M istotally σ –artinian.(b) M / T istotally σ –artinian. 10t is a direct consequence of Proposition (1.4.), since every totally σ –torsion module istotally σ –artinian.In the same line, we find that finitely cogenerated modules have their own characteriza-tion. The following is Theorem 3.4 in [ ] . Theorem 3.4.
Let M bean A –module,thefollowingstatements areequivalent:(a) M isfinitelycogenerated.(b) M isstronglytotally σ A \ p –finitelycogenerated,forevery p ∈ Spec ( A ) .(c) M isstronglytotally σ A \ m –finitelycogenerated,forevery m ∈ Max ( A ) .P ROOF (a) ⇒ (b) ⇒ (c). They are obvious.(c) ⇒ (a). Let { N i | i ∈ I } be a family of submodules such that ∩ i ∈ I N i =
0, for ev-ery maximal ideal m ⊆ A there exist a finite subset J m ⊆ I and s m ∈ A \ m such that ( ∩ j ∈ J m N j ) s m =
0. If h = 〈 s m | m ∈ Max ( A ) 〉 6 = A there exists a maximal ideal m suchthat h ⊆ m , which is a contradiction. Therefore, h = A , and there are maximal ideals m , . . . , m t such that 〈 s m , . . . , s m t 〉 = A . We define J = ∪ ti = J m i ⊆ I , which is finite, andsatisfies ( ∩ j ∈ J N j ) s m i =
0, for every i =
1, . . . , t . Hence ∩ j ∈ J N j = ( ∩ j ∈ J N j ) 〈 s m , . . . , s m t 〉 = M is finitely cogenerated. ƒ Also, if A = D is an integral domain strongly totally σ –finitely cogenerated then D is afield, whenever σ = Proposition 3.5. If D isa σ –torsionfreestronglytotally σ –finitely cogenerated integraldomain,then D isafield. The conversealways holds.P ROOF
We claim 0 ⊆ D is strongly prime, see [ ] . Indeed, let { a i | i ∈ I } be a familyof ideals such that ∩ i ∈ I a i =
0. By the hypothesis, there exist J ⊆ I , finite, and h ∈ L ( σ ) such that ( ∩ j ∈ J a j ) h =
0. Since A is σ –torsionfree, then ∩ j ∈ J a j =
0. Otherwise, since0 ⊆ D is prime, there exists j ∈ J such that a j = D contains aminimum non-zero ideal, say a = aD . For any 0 = x ∈ D we have a x =
0, and aD ⊆ a x D ,this means that there exists y ∈ D such that a = a x y , and 1 = x y , hence x is invertible. ƒ We put the condition that D is σ –torsionfree only to avoid the trivial case in which σ = Let p ⊆ A be a prime ideal, we consider σ A \ p , the hereditary torsion theory cogeneratedby A / p , or equivalently, the hereditary torsion theory generated by the multiplicativelysubset A \ p . For every torsion theory σ we associate the following sets of ideals:111) L ( σ ) , the Gabriel filter of σ .(2) Z ( σ ) = L ( σ ) ∩ Spec ( A ) . In particular, if p ⊆ q are prime ideals and p ∈ Z ( σ ) , then q ∈ Z ( σ ) .(3) C ( A , σ ) = { a | A / a ∈ F σ } .(4) K ( σ ) = C ( A , σ ) ∩ Spec ( A ) , it is the complement of Z ( σ ) in Spec ( A ) . In particular,if p ⊆ q are prime ideals and q ∈ Z ( σ ) , then p ∈ Z ( σ ) .(5) C ( σ ) = Max K ( σ ) .If σ is of finite type, then σ = ∧{ σ A \ p | p ∈ K ( σ ) } . Otherwise, σ = ∧{ σ A \ p | p ∈ C ( σ ) } whenever A is σ –noetherian, because σ A \ q ≤ σ A \ p whenever p ⊆ q , for any prime ideals p , q . Proposition 4.1.
Let p ⊆ A be a primeideal. If A is totally σ A \ p –artinian, then p ⊆ A isaminimal primeideal.P ROOF
Let q $ p be prime ideals such that A is totally σ A \ p –artinian. Taking the quotientby the ideal q , p : A −→ A / q , we have that A / q is a totally p ( σ A \ p ) –artinian domain, hence p ( σ A \ p ) is the usual hereditary torsion theory in a domain, i.e., L ( p ( σ A \ p )) contains onlythe non–zero ideals of A / q . Therefore, q / ∈ L ( σ A \ p ) , which is a contradiction. ƒ Corollary 4.2.
Let A be a totally σ –artinian ring, every prime ideal p ∈ K ( σ ) is a min-imal prime ideal. In consequence, K ( σ ) = C ( σ ) , i.e., every prime ideal in K ( σ ) ismaximal in K ( σ ) .P ROOF
Let p ∈ K ( σ ) , then σ ≤ σ A \ p , and A is σ A \ p –artinian. Therefore, p is a minimalprime ideal, hence maximal in K ( σ ) . ƒ For any multiplicatively closed subset Σ ⊆ A , and σ = σ Σ , we obtain Proposition 2.5 in [ ] .Since every totally σ –artinian ring is σ –artinian, we establish the next result for σ –artinian rings. Corollary 4.3.
Let A bea σ –artinian ring,the followingstatementshold:(1) C ( A , σ ) isartinian, and theconverse also holds.(2) A is σ –noetherian. Inparticular, σ isoffinite type.(3) K ( σ ) isafinite set,say K ( σ ) = { p , . . . , p t } .(4) Thereexistsamultiplicativelyclosedsubset Σ ⊆ A suchthat L ( σ ) = { a | a ∩ Σ = ∅ } and A Σ isartinian. The conversealso holds.P ROOF (1). It is just the definition.(2). It is Hopkins’ Theorem. See [ ] .(3). If K ( σ ) is not finite, there exists a numerable family of prime ideals { p n | n ∈ N } ⊆K ( σ ) . Then we may build a decreasing chain of ideals p ⊇ p p ⊇ p p p ⊇ · · · . By the12ypothesis this chain stabilizes, and there exists an index m such that Cl A σ ( p · · · p m ) = Cl A σ ( p · · · p m p m + ) = · · · . Therefore, for any x ∈ p · · · p m there exists h ∈ L ( σ ) suchthat x h ⊆ p · · · p m p m + ⊆ p m + . Since h * p m + , we have x ∈ p m + . In consequence, p · · · p m ⊆ p m + , and there exists an index i such that p i ⊆ p m + , which is a contradiction.(4). See [
1, Corollary 7.24 ] . ƒ The following result appears as Theorem 2.2 in [ ] . Theorem 4.4.
Let A be aring,thefollowingstatements are equivalent:(a) A isartinian.(b) A istotally σ A \ p –artinian,forevery p ∈ Spec A .(c) A istotally σ A \ m –artinian,forevery m ∈ Max ( A ) .P ROOF (a) ⇒ (b) ⇒ (c). It is evident.(c) ⇒ (a). Let a ⊇ a ⊇ · · · be a decreasing chain of ideals. For every maximal ideal m there exist m m and s m ∈ A \ m such that a m m ⊆ a s for every s ≥ m m . Let G = { s m | m ∈ Max ( A ) } . If ( G ) = A , there exists a maximal ideal m such that ( G ) ⊆ m , which is acontradiction. Thus we have ( G ) = A , and there exist s , . . . , s t ∈ G such that P ti = s i A = A .Let s i = s m i , and m i = m m i for every i =
1, . . . , t . If m = max { m , . . . , m t } we have a m ⊆ a m i , then a m s i ⊆ a m i s i ⊆ a s ,for every s ≥ m . In consequence, a m = a m P ti = s i A ⊆ a s for every s ≥ m , and the chainstabilizes. ƒ We may study this result in order to characterize totally σ –artinian rings. Theorem 4.5.
Let A be a ring and σ be a finite type hereditary torsion theory. Thefollowingstatementsare equivalent:(a) A istotally σ –artinian.(b) K ( σ ) = C ( σ ) isfiniteand A istotally σ A \ p –artinianforeveryprimeideal p ∈ C ( σ ) .P ROOF (a) ⇒ (b). See Corollaries (4.2.) and (4.3.).(b) ⇒ (a). Let a ⊇ a ⊇ · · · be a chain of ideals. For every p ∈ C ( σ ) there exists m p ∈ N and h p ∈ L ( σ A \ p ) such that a m p h p ⊆ a s for every s ≥ m p . If we take m = max { m p | p ∈C ( σ ) } , and h = Q { h p | p ∈ C ( σ ) } , then mh ⊆ a s for every s ≥ m , which shows that A istotally σ –artinian. ƒ Simple and maximal modules
In order to show that every totally σ –artinian ring is totally σ –noetherian we need tostudy simple modules and maximal ideal relative to the hereditary torsion theory σ .We shall use the preceding studies of minimal and maximal elements as appears in sec-tions(3) and (6), respectively. The example of σ –torsion. The classical theory Let M be an A –module. We have that M is σ –artinian if, and only if, the family C ( M , σ ) of σ –closed submodules M satisfies the decreasing chain condition, or equivalently theminimal condition, which are also equivalent to the condition that for every decreasingchain N ⊇ N ⊇ · · · of submodules there exists m ∈ N such that Cl M σ ( N m ) = Cl M σ ( N s ) , forevery s ≥ m .A submodule N ⊆ M is σ –minimal if Cl M σ ( N ) ⊆ M is a minimal element in C ( M , σ ) \{ σ M } , or equivalently if C ( N , σ ) = { σ N , N } . This means that N is not σ –torsion, and forevery submodule L ⊆ N we have either L is σ –torsion or L is not σ –torsion, and in thiscase L ⊆ σ N , it is σ –dense. Observe that if M is not σ –torsion, a submodule N ⊆ M is σ –minimal if, and only if, N is a minimal in the family M = { N ⊆ M | N is not σ –torsion } .An A –module M is σ –simple if C ( M , σ ) = { σ M , M } . If, in addition, M is σ –torsionfree,we name M a σ –cocritical A –module.We may dualize σ –artinian to obtain σ –noetherian modules. In the case of σ –maximalsubmodules, we have that N ⊆ M is σ –maximal if M / N is not σ –torsion, and for everysubmodule N ⊆ L ⊆ M such that M / L is not σ –torsion we have N ⊆ σ L , which isequivalent to say that M / N is a σ –simple A –module. Observe that if M is not σ –torsion,a submodule N ⊆ M is σ –maximal if, and only if, N is a maximal the family M = { N ⊆ M | M / N is not σ –torsion } .A submodule N ⊆ M is called σ –critical if it is σ –maximal and M / N is σ –torsionfree. The example of totally σ –torsion. Simple modules When we study modules and the totally σ –torsion we need to change the paradigm.Thus, let M be a totally σ –artinian A –module, and consider the family of submodules M = { N ⊆ M | N is not totally σ –torsion } . M is not empty whenever M is not totally σ –torsion. If M is totally σ –artinian, thereexists a σ –minimal element in M , say N , that satisfies:(1) N is not totally σ –torsion, 142) There exists h ∈ L ( σ ) such that for every H ⊆ N , which is not totally σ –torsion, wehave N h ⊆ H .In general, for any A –module M , a submodule N satisfying (1) and (2) is called a totally σ –minimal submodule of M . An A –module M is called totally σ –simple whenever M is a σ –minimal element of M , i.e.,(1) M is not totally σ –torsion and(2) there exists h ∈ L ( σ ) such that for every not totally σ –torsion submodule H ⊆ M we have M h ⊆ H .Let M be a totally σ –simple A –module with companion ideal h ∈ L ( σ ) , i.e., h satisfiesthat for every H ⊆ M which is non totally σ –torsion we have M h ⊆ H .Observe that we have: Proposition 5.1.
Let M beatotally σ –simple A –modulewithcompanionideal h ∈ L ( σ ) ,the followingstatementshold:(1) M h is not totally σ –torsion, hence M h ⊆ M is the minimum of all not totally σ –torsion submodules of M . In particular, every not totally σ –torsion of M is totally σ –simple.(2) M h isalso totally σ –simple,and everypropersubmodule iftotally σ –torsion.(3) For any ideal h ′ ∈ L ( σ ) we always have M h ⊆ M h ′ . In particular, if h ′ ⊆ h then M h = M h ′ .(4) If h ′ ∈ L ( σ ) isanotherideal companionto M ,then M h ′ = M h .(5) Let M ′ be a totally σ –simple A –module and f : M −→ M ′ be a surjective map withkernel K ,then K istotally σ –torsion.P ROOF (1) to (4) are straightforward.(5). If K ⊆ M is not totally σ –torsion, and h ∈ L ( σ ) is the companion ideal of M , then M h ⊆ K , and we have M ′ h =
0, which is a contradiction. ƒ If M is totally σ –simple with companion ideal h , we call M h the core submodule of M . Proposition 5.2. If M ⊆ M ′ satisfies that there exists h ∈ L ( σ ) such that M ′ h ⊆ M ,then M is totally σ –simple if, and only if, M ′ is. In addition, the core of M and M ′ areequal.P ROOF If M is totally σ –simple, h ∈ L ( σ ) is a companion ideal, and H ⊆ M ′ is nottotally σ –torsion, then H h ⊆ M is not totally σ –torsion, hence M h ⊆ H h ⊆ H , and M ′ h h ⊆ H . Otherwise, if M ′ is totally σ –simple, h ′ ∈ L ( σ ) is a companion ideal, and H ⊆ M is not totally σ –torsion, since H ⊆ M ′ , we have M h ′ ⊆ M ′ h ′ ⊆ H , and M is totally σ –simple.If M ⊆ M ′ are totally σ –simple modules we may assume h is the same companion idealto M and M ′ , then we have M h = M ′ h . Indeed, M ′ h ⊆ M , and M ′ h = M ′ hh ⊆ M h ⊆ M ′ h .The core of M / L is a quotient of the core of M . ƒ roposition 5.3. Let L ⊆ M be a totally σ –torsion submodule, then M is totally σ –simple if, and only if, M / L is. In this case, the core of M / L is a quotient of the core of M .P ROOF If M is totally σ –simple with companion ideal h ∈ L ( σ ) , and H / L ⊆ M / L is anot totally σ –torsion submodule, then H ⊆ M is not totally σ –torsion, hence M h ⊆ H ,and ( M / L ) h = ( M h + L ) / L ⊆ H / L . Otherwise, if M / L is totally σ –simple with companionideal h ∈ L ( σ ) , and H ⊆ M is not totally σ –torsion, then ( H + L ) / L is not totally σ –torsion, hence ( M / L ) h ⊆ ( H + L ) / L , and M h + L ⊆ H + L . Since L is totally σ –torsion,there exists h ′ ∈ L ( σ ) such that L h ′ =
0. Therefore, M hh ′ ⊆ H h ′ ⊆ H , and M is totally σ –simple.If M h is the core of M , its image is M h / ( M h ∩ L ) , which is totally σ –simple and everyproper submodule is totally σ –torsion. Indeed, if H / ( M h ∩ L ) $ M h / ( M h ∩ L ) , then H $ M , so it is totally σ –torsion, hence H / ( M h ∩ L ) is. ƒ An A –module M is core totally σ –simple whenever M is the core of a totally σ –simplemodule, i.e., whenever M is not totally σ –torsion and every proper submodule is totally σ –torsion.Observe that if M is core totally σ –simple, then M h = M for every h ∈ L ( σ ) . Indeed,if h ∈ L ( σ ) is the companion ideal of M , then M h = M , and for every h ∈ L ( σ ) wehave: M h = M hh = M .If M is a core totally σ –simple A –module, we have two different cases:(1) σ M = M . In this case, σ M is totally σ –torsion and M /σ M is σ –torsionfree andcore totally σ –simple, hence it is simple. Indeed, every proper submodule is totally σ –torsion, hence zero.(2) σ M = M . In this case, since every proper quotient is σ –torsion, it follows that M has no simple quotients. In particular, M is not finitely generated.Let M be a core totally σ –simple A –module, we have the following two possibilities:(1) M is not cyclic. Since every proper submodule of M is totally σ –torsion, we have M is σ –torsion. Hence M is not finitely generated because it is not totally σ –torsion.(2) M is cyclic. Since it is not totally σ –torsion, then σ M = M , and M has simple σ –torsionfree quotients. Otherwise, every simple quotient of M is σ –torsionfree.Consequences of this fact are: σ ( M ) is totally σ –torsion, and σ M is contained inthe intersection of all maximal submodules of M . Lemma 5.4.
Let A be a ring and a ⊆ A be a proper ideal, then M = A / a is core totally σ –simple if,and onlyif, itsatisfies:(1) a / ∈ L ( σ ) ,(2) a + h = A ,forevery h ∈ L ( σ ) ,and(3) ( a : b ) ∈ L ( σ ) ,foreveryproperideal b ⊇ a .16 ROOF If A / a is core totally σ –simple, then it is not totally σ –torsion, i.e., a / ∈ L ( σ ) .For any h ∈ L ( σ ) we have ( A / a ) h = A / a , i.e., a + h = A . Since every proper submoduleof A / a is totally σ –torsion, then for every b ⊇ a we have ( a : b ) ∈ L ( σ ) . ƒ In particular, for any maximal ideal m the simple A –module A / m is totally σ –simple if,and only if, m / ∈ L ( σ ) . Lemma 5.5.
The class M is σ –lowerclosed.P ROOF
Indeed, if H ⊆ M and there are N ∈ M and h ∈ L ( σ ) such that N h ⊆ H , then H is not totally σ –torsion. On the contrary there exists h ′ ∈ L ( σ ) such that H h ′ =
0, hence H hh ′ =
0, and N is totally σ –torsion. ƒ If M is totally σ –artinian, there are minimal elements in M ; every minimal element N of M is a core totally σ –simple module, i.e., it satisfies:(1) N is not totally σ –torsion, and(2) Every proper submodule of N is totally σ –torsion. Lemma 5.6.
Let M beatotally σ –simple module,forany submodule N ⊆ M wehave:(1) If N isnot totally σ –torsion,then N istotally σ –simple. Inaddition, N and M havethesame core.(2) The quotient M / N is either totally σ –torsion whenever N is non totally σ –torsion,ortotally σ –simplewhenever N istotally σ –torsion.P ROOF (1) is straightforward.(2). If N is non totally σ –torsion, there exists h ∈ L ( σ ) such that M h ⊆ N , hence M / N is totally σ –torsion. Otherwise, if N is totally σ –torsion, then M / N is non totally σ –torsion; on the other hand, for any non totally σ –torsion submodule L / N ⊆ M / N ,since L ⊆ M is non totally σ –torsion, there exists h ∈ L ( σ ) such that M h ⊆ L ; hence ( M / N ) h ⊆ ( L / N ) , and M / N is totally σ –simple. ƒ Lemma 5.7. If N , N aretotally σ –simplemodules,and f : N −→ N isamodulemap,then f iseitherzeroorsurjective. In this section we assume the reader knows about totally σ –noetherian rings and modulesas it was exposed in [ ] .Let M be a totally σ –noetherian A –module, and consider the family of submodules M ′ = { N ⊆ M | M / N is not totally σ –torsion } .We have that M ′ is nonempty whenever M is not totally σ –torsion.17 emma 6.1. Let M beanontotally σ –torsionmodule,theclass M ′ is σ –upper closed.P ROOF If H ⊆ M and there are N ∈ M ′ and h ∈ L ( σ ) such that H h ⊆ N , then H ∈ M ′ .We show that M / H is not totally σ –torsion. On the contrary, there exists h ∈ L ( σ ) such that M h ⊆ H ; hence M h h ⊆ H h ⊆ N , and M / N is totally σ –torsion, which is acontradiction. ƒ Since M is totally σ –noetherian, there are maximal elements in M ; a maximal elementof M ′ is called a totally σ –maximal submodule of M . We define a submodule N of M tobe a core totally σ –maximal submodule whenever it satisfies:(1) M / N is not totally σ –torsion.(2) N is maximal in the σ –upper closed family N ′ = { L ⊆ M | N ⊆ L , M / L is not totally σ –torsion } .In the same way, we can define a totally σ –maximal submodule of an A –module M whenever(1) M / N is not totally σ –torsion, and(2) N is σ –maximal in N ′ , i.e., there exists h ∈ L ( σ ) such that for every H ∈ N ′ suchthat N ⊆ H we have H h ⊆ N .Even, we can dualize the notion of totally σ –simple submodule, in defining a submodule N ⊆ M to be totally σ –cosimple if it satisfies:(1) M / N is not totally σ –torsion, and(2) There exists h ∈ L ( σ ) such that for every N ⊆ H $ M satisfying that H / N is nottotally σ –torsion we have M h ⊆ H .and defining core totally σ –cosimple if, in addition, for every N ⊆ H $ M we have that H / N is totally σ –torsion. Lemma 6.2.
Let M be a core totally σ –simple A –module, then Ann ( M ) ⊆ A is a coretotally σ –cosimpleideal.P ROOF
Since M is core totally σ –simple we have two possibilities for M :(1) σ M = M , and(2) σ M = M .In case (2) there exists x ∈ M \ σ M , such that x A = M . Indeed, since x A ⊆ M and itis not totally σ –torsion, then x a = M . Hence Ann ( M ) = Ann ( x ) , and A / Ann ( x ) is nottotally σ –torsion. The rest is obvious.In case (1) we have that M is cyclic and we can proceed in the same way. ƒ roposition 6.3. Let M be a totally σ –simple A –module, then Ann ( M ) ⊆ A is totally σ –cosimple.P ROOF
Let M h ⊆ M the core of M , we have Ann ( M h ) = Ann ( M ) : h , hence we can builda short exact sequence 0 → Ann ( M ) : h Ann ( M ) → A Ann ( M ) → A Ann ( M h ) →
0. Since Ann ( M ) : h Ann ( M ) is totally σ –torsion, we have the result. ƒ If M = A we may determine more precisely the core totally σ –cosimple ideals. Proposition 6.4.
Let a ⊆ A bean ideal. If a ⊆ A iscoretotally σ –cosimplethen:(1) a isprime.(2) a ismaximalin K ( σ ) ,i.e., a ∈ C ( σ ) .Inconclusion,coretotally σ –cosimpleidealsare exactlythe idealsin C ( σ ) .P ROOF
Since a ⊆ A is core totally σ –cosimple, it is not totally σ –torsion, hence a / ∈ L ( σ ) .(1). Let a , a ⊆ A be proper ideals properly containing a such that a a ⊆ a . Since a $ a $ A , then A / a is totally σ –torsion, hence a ∈ L ( σ ) . Similar result holds for a .Therefore, a a ∈ L ( σ ) , and a ∈ L ( σ ) , which is a contradiction.(2). Since a / ∈ L ( σ ) is prime, then a ∈ K ( σ ) . Let a ⊆ p ∈ K ( σ ) , since A / p is σ –torsionfree and non–zero, it is not totally σ –torsion, hence p = a . Therefore, a ∈ K ( σ ) is maximal and a ∈ C ( σ ) .The converse is obvious because for any p ∈ C ( σ ) we have that A / p is σ –cocritical. ƒ The core totally σ –Jacobson radical of an A –module M is defined as the intersection ofall core totally σ –cosimple submodule, and we represent it by t Jac ( M ) . Lemma 6.5.
Let M bean A –module,then t Jac ( M / t Jac ( M )) = ROOF
We have N ⊆ M / t Jac ( M ) is core totally σ –cosimple if, and only if, N ⊆ M iscore totally σ –cosimple. ƒ Proposition 6.6. (1) Everytotally σ –simple A –moduleistotally σ –artinian.(2) Everycoretotally σ –simple A –moduleistotally σ –noetherian.(3) Everytotally σ –simple A –moduleistotally σ –noetherian.P ROOF (1). Let M be totally σ –simple and N ⊇ N ⊇ · · · be a decreasing chain ofsubmodules of M . If for every index i we have that N i is not totally σ –torsion, and h ∈ L ( σ ) is the companion ideal of M , then M h ⊆ N i , hence N h ⊆ M h ⊆ N i , for everyindex i . If there exists an index m such that N m is totally σ –torsion, there exists h ′ ∈ L ( σ ) such that N m h ′ = ⊆ N s , for every s ≥ m .(2). If N ⊆ N ⊆ · · · is an ascending chain of submodules of M , and M is core totally σ –simple we studied the two cases in page 16.192.1). If M is σ –torsion, then M is cyclic, hence ∪ n N n ⊆ M is either totally σ –torsion or ∪ n N n = N . In the second case, there exists an index m such that M = N m . In both casesthe chain is σ –stable.(2.2). If M is not σ –torsion, then σ M is totally σ –torsion. If ∪ n N n ⊆ σ M , then thechain σ -stabilizes. If there exists an index m such that N m * σ M , and h ∈ L ( σ ) is thecompanion ideal of M , then M h ⊆ N m , and in this case the chain stabilizes.(3). If M is totally σ –simple, with companion ideal h ∈ L ( σ ) , then M h is core totally σ –simple, hence it is totally σ –noetherian. Otherwise, M / M h is totally σ –torsion, hencetotally σ –noetherian. Therefore, M is totally σ –noetherian because it is an extension of M h by M / M h . ƒ Lemma 6.7.
Let M beanartinian A –modulesuchthat t Jac ( M ) = T ⊆ M ,totally σ –torsion such that M / T ⊆ ⊕ tj = S j , for a finite family of core totally σ –simple A –modules. Inparticular, M istotally σ –noetherian.P ROOF If t Jac ( M ) = σ –cosimple submodules iszero, and since M is σ –finitely cogenerated, there exists a finite family of core totally σ –cosimple submodules, { N , . . . , N t } such that ∩ tj = N j is totally σ –torsion. If we call T = ∩ tj = N i , then M / T is a submodule of ⊕ tj = ( M / N j ) . Finally, since ⊕ tj = ( M / N j ) istotally σ –noetherian, then M / T is totally σ –noetherian, and we have M is totally σ –noetherian. ƒ Theorem 6.8. If A isatotally σ –artinian ring,then A istotally σ –noetherian.P ROOF
Let J = t Jac ( A ) . If A is totally σ –artinian then A / J is totally σ –artinian andtotally σ –noetherian, by Lemma (6.7.). We consider A / J , because for every ideal a ⊆ A we have that a ⊆ A is a core totally σ –cosimple ideal if, and only if, a / J ⊆ A / J is coretotally σ –cosimple, then t Jac ( A / J ) = J / J , and the same holds for every m ∈ N , i.e, t Jac ( A / J m ) = J / J m .The decreasing chain J = J ⊇ J ⊇ · · · is σ –stable, hence there exist m ∈ N and h ∈L ( σ ) such that J m h ⊆ J s for every s ≥ m . We do induction on m . Let us assume m = J / J is totally σ –torsion, and we have a short exact sequence0 −→ J / J −→ A / J −→ A / J −→ J / J and A / J are totally σ –artinian and totally σ –noetherian, then A / J is. Weassume the result holds for any positive integral number smallest than m and that J m h ⊆ J s for every s ≥ m . Consider the short exact sequence0 −→ J m / J m + −→ A / J m + −→ A / J m −→ J m / J m + is totally σ –torsion and A / J m is totally σ –artinian and totally σ –noethe-rian then A / J m + is. ƒ For any ring A the totally Jacobson σ –radical of A which is the intersection of all core to-tally σ –cosimple ideals is the intersection of all elements in C ( σ ) , see Proposition (6.4.),which coincides with the Jacobson σ –radical of A , i.e., t Jac ( A ) = Jac σ ( A ) = ∩{ p | p ∈ C ( σ ) } .We show that there exist enough core totally σ –cosimple submodule in the followingsense. Lemma 6.9.
Let σ be a finite type hereditary torsion theory, for any totally σ –finitelygenerated module M and any proper submodule N ⊆ M such that M / N is not totally σ –torsion,thereexistsacoretotally σ –cosimplesubmodule N ⊆ H $ M .P ROOF
Let Γ = { H ⊆ M | N ⊆ H ⊆ M and M / H is not totally σ –torsion } . If Γ = { N } ,then N is maximal among those submodules which are not totally σ –torsion, hence it iscore totally σ –cosimple. Otherwise, for any increasing chain N ⊆ N ⊆ · · · in Γ , we con-sider ∪ n ≥ N n . If M / ∪ n ≥ N n is totally σ –torsion, there exists h ∈ L ( σ ) , finitely generated,such that M h ⊆ ∪ n ≥ N n , and there is an index m such that M h ⊆ N m . Otherwise, thereexist N ′ ⊆ M , finitely generated, and h ′ ∈ L ( σ ) , finitely generated, such that M h ′ ⊆ N ′ .In consequence, M h ′ h ⊆ N h ⊆ M h ⊆ ∪ n ≥ N n , and there exists an index m such that M h ′ h ⊆ N m , which is a contradiction. Thus, Γ is an inductive set of submodules, and byZorn’s lemma we have that Γ has maximal submodules. A maximal submodule N ⊆ M in Γ is a core totally σ –cosimple submodule. ƒ Of particular interest is the case in which M = A ; in this case for every ideal a ⊆ A suchthat a / ∈ L ( σ ) , there exist p ∈ C ( σ ) such that a ⊆ p . References [ ] T. Albu and C. Nastasescu,
Relative finiteness in module theory , Pure and AppliedMathematics. A Series of Monographs and Textbooks, 84, Marcel Dekker, New York,1984. [ ] D. D. Anderson and T. Dumitrescu,
S-noetherian rings , Comm. Algebra (2002), 4407–4416. [ ] M. Eljeri,
S–strongly finite type rings , Asian Research J. Math. (2018), 1–9.21 ] J. S. Golan,
Torsion theories , Pitman Monographs and Surveys in Pure and AppliedMath. vol. 29, Pitman, 1986. [ ] C. Gottlieb,
On strongly prime ideals and strongly zero-dimensional rings , J. AlJ. Al-gebra.
16 (10) (2017), 9 pages. [ ] E. Hamann, E. Houston, and J. L. Johnson,
Properties of upper to zero in R [ X ] , PacificJ. Math. (1988), 65–79. [ ] Ahmed Hamed,
S-noetherian spectrum condition , Comm. Algebra
46 (8) (2018),3314–3321. [ ] P. Jara,
An extension of S–noetherian rings and modules , University of Granada(2020), 16 pp. arXiv: 2011.03008. [ ] A. V. Jategaonkar,
Endomorphism rings of torsionless modules , Trans. Amer. Math.soc. (1971), 457–466. [ ] Jung Wook Lim,
A note on S–noetherian domains , Kyungpook Math. J. (2015),507–514. [ ] Jung Wook Lim and Dong Yeol Oh,
S-noetherian properties on amalgamated algebraalong an ideal , J. Pure Appl. Algebra (2014), 1075–1080. [ ] E. S. Sevim, U. Tekir, and S. Koc,
S-artinian rings and finitely S-cogenerated rings , J.Algebra Appl. xx (2020), 16 pages. [ ] B. Stenström,
Rings of quotients , Springer–Verlag, Berlin, 1975. [ ] L Zhongkui,
On S–noetherian rings , Arch. Math. (Brno)43