Left S-module Research Articles

On essential-prime sub-modules

Let S be a commutative ring with identity, let A be an until (left) S-module. In this paper, we study the concepts : essential prime sub-modules as a generalization of prime sub-modules. A sub-module H of an S-module A is said to be essential-prime if whenever ri ∈ Si, a ∈ A with and rai ∈ Hi theni eitheri a ∈ Hi or ri ∈ [H : iA]. We give a number of basic properties and characterizations for this type of sub-modules.

FP n -injective and FP n -flat modules with respect to a semidualizing bimodule

Let S and R be rings, be a semidualizing bimodule and be an integer or We introduce and study C-FP n -injective and C-FP n -flat modules as a common generalization of some known modules such as C-injective (resp. C-FP-injective, C-weak injective) and C-flat (resp. C-projective, C-weak flat) modules. Suppose that is a faithfully semidualizing bimodule. We give some equivalent characterizations of left n-coherent rings in terms of C-FP n -flat left S-modules and C-FP n -injective left R-modules. Then we show that the pairs and are coproduct-closed and product-closed duality pairs and both and are covering and preenveloping, where and denote the classes of C-FP n -flat left S-modules and C-FP n -injective left R-modules respectively. Finally, we investigate Foxby equivalence relative to C-FP n -injective and C-FP n -flat modules.

Special precovering classes in comma categories

Let T be a right exact functor from an abelian category ℬ into another abelian category \({\mathscr A}\). Then there exists a functor p from the product category \({\mathscr A} \times {\mathscr B}\) to the comma category (\(\left( {T \downarrow {\mathscr A}} \right)\)). In this paper, we study the property of the extension closure of some classes of objects in (\(\left( {T \downarrow {\mathscr A}} \right)\)), the exactness of the functor p and the detailed description of orthogonal classes of a given class \({\rm{P}}\left( {{\cal X},{\cal Y}} \right)\) in (\(\left( {T \downarrow {\mathscr A}} \right)\)). Moreover, we characterize when special precovering classes in abelian categories \({\mathscr A}\) and ℬ can induce special precovering classes in (\(\left( {T \downarrow {\mathscr A}} \right)\)). As an application, we prove that under suitable cases, the class of Gorenstein projective left Λ-modules over a triangular matrix ring \({\rm{\Lambda }} = \left( {\matrix{ R & M \cr 0 & S \cr } } \right)\) is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering. Consequently, we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.

Frobenius functors and Gorenstein flat dimensions

We prove that if the Frobenius functor F (from the category of left R-modules to the category of left S-modules) is faithful, then for any R-module X, the Gorenstein flat dimension of X is equal to the Gorenstein flat dimension of F(X), which is motivated by a result of Nakayama and Tsuzuku about relations between Frobenius extensions and flat dimension of modules. It is a well-known metatheorem of Holm that every result in homological algebra has a counterpart in Gorenstein homological algebra. Note that the flat dimension of X can be different from that of F(X). Our main result provides a counterexample to the converse of Holm’s metatheorem. In addition, some applications are given.

A note on the stability of pure-injective modules

ABSTRACTLet R be a ring. Then a left R-module N is pure-injective if and only if HomR(M,N) is a pure-injective left S-module for any ring S and any (R,S)-bimodule . If R is a commutative ring and M,N are R-modules with N pure-injective, then is a pure-injective R-module for any n≥0. Let R and S be rings and let be an (S,R)-bimodule and M a finitely presented left R-module. If N is pure-injective as a left S-module, then the left S-module N⊗RM is pure-injective; and if R is left coherent, then the left S-module is pure-injective for any n≥1.

Gorenstein Modules and Dualizing Modules

In this article, we study the characterizations of Gorenstein injective left S-modules and finitely generated Gorenstein projective left R-modules when there is a dualizing S-R-bimodule associated with a right noetherian ring R and a left noetherian ring S.

A characterization of Auslander category

In this paper, we discuss the Bass class and the Auslander class with respect to a semidualizing module over an associative ring. Let _SC_R be a semidualizing module, we proved that the Bass class B_C (R) is a right orthogonal subcategory of some right R-module; and that the Auslander class A_C (S) is a left orthogonal subcategory of the character module of some left S-module. As an application, we introduce the notion of the minimal semidualizing module, and get a one to one correspondence between the isomorphism classes of minimal semidualizing R-modules and maximal classes among coresolving preenvelope classes of Mod R with the same Ext-projective generators in gen^* R.

ALMOST PRINCIPALLY SMALL INJECTIVE RINGS

Let R be a ring and M a right R-module, S = <TEX>$End_R$</TEX>(M). The module M is called almost principally small injective (or APS-injective for short) if, for any a <TEX>${\in}$</TEX> J(R), there exists an S-submodule <TEX>$X_a$</TEX> of M such that <TEX>$l_Mr_R$</TEX>(a) = Ma <TEX>$Ma{\bigoplus}X_a$</TEX> as left S-modules. If <TEX>$R_R$</TEX> is a APS-injective module, then we call R a right APS-injective ring. We develop, in this paper, APS-injective rings as a generalization of PS-injective rings and AP-injective rings. Many examples of APS-injective rings are listed. We also extend some results on PS-injective rings and AP-injective rings to APS-injective rings.

GPQ modules and generalized Armendariz modules

Let M_R be a right R-module. We introduce the concept of right generalized p.q.-Baer modules (or simply, right GPQ modules) to extend the notion of right p.q.-Baer modules. We study on the relationship between the GPQ property of a module M_R and various quasi-Armendariz properties. We prove that every right GPQ module is a quasi-Armendariz module. As a sequence, we obtain a general form of some known results considering the p.q.Baer property of a ring, some known results are extended. Moreover, we prove that for the formal triangular ring R constructed from a pair of rings S, T and a bimodule _SM_T, R is weak Armendariz if and only if (1) S and T are weak Armendariz rings. (2) _SM and M_T are weak Armendariz as a left S-module and right R-module. (3) If s(x)s'(x)=t(x)t'(x)=0, then s(x)M[x] \cap M[x]t'(x)=0. This gives the relationship of weak Armendarizness between R and S, T, _SM_T, which plays a very important role in ring theory.

ON COHERENCE OF ENDOMORPHISM RINGS

AbstractLet R be a ring and U a left R-module with S=End(RU). The aim of this paper is to characterize when S is coherent. We first show that a left R-module F is TU-flat if and only if HomR(U,F) is a flat left S-module. This removes the unnecessary hypothesis that U is Σ-quasiprojective from Proposition 2.7 of Gomez Pardo and Hernandez [‘Coherence of endomorphism rings’, Arch. Math. (Basel)48(1) (1987), 40–52]. Then it is shown that S is a right coherent ring if and only if all direct products of TU-flat left R-modules are TU-flat if and only if all direct products of copies of RU are TU-flat. Finally, we prove that every left R-module is TU-flat if and only if S is right coherent with wD(S)≤2 and US is FP-injective.

On a Natural Duality Between Grothendieck Categories

A right R-module M with endomorphism ring S is called a costar module if it induces the duality between the class of M R -torsionless right R-modules X with Hom R (X, M R ) finitely generated over S and the class of S M-torsionless finitely generated left S-modules. In this article we consider, more generally, a pair of additive, contravariant functors—which are adjoint on the right—between Grothendieck categories, and describe a natural duality induced by them. Our result subsumes the situation mentioned above but also, for example, a rigid graded duality that gives the notion of graded costar module.

Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical. In addition, we prove that if R → S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules, then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical. We also give some applications of these results.

Change of ring and torsion-theoretic injectivity

Let τ be a hereditary torsion theory in R-Mod. Then any ring homomorphism γ: R → S induces in S-Mod a torsion theory σ given by the condition that a left S-module is σ-torsion if and only if it is τ-torsion as a left R-module. We show that if γ: R → S is a ring epimorphism and A is a τ-injective left R-module, then AnnA Ker(γ) is σ-injective as a left S-module. As a consequence, we relate τ-injectivity and σ-injectivity, and we give some applications.

Krull dimension of bimodules

Let R and S be rings and let M be a left S-, right R-bimodule such that M has Krull dimension both as a left S-module and as a right R-module and M / N has the same Krull dimension as a left S- and as a right R-module for every sub-bimodule N of M. Suppose that M is a finitely generated left S-module. Then it is proved that the Krull dimension of the right R-module M is the Krull dimension of a k-critical right R-module M / K for some prime submodule K of the right R-module M.

ON THE COMPLEMENTARITY OF TWO QUASI-TILTING TRIPLES

Let R and S be two rings. Each category equivalence between a torsion class of left (right) R-modules and a torsion-free class of left (right) S-modules is represented by a left (right) quasi-tilting triple. Suppose we have a pair of equivalences T ⇄ Y and X F between the torsion class T of R-modules and the torsion-free class Y of S-modules and between the torsion class X of S-modules and the torsion-free class F of R-modules. Denote by (R, V, S) and (S, U, R) the quasi-tilting triples representing these equivalences. We say that (R, V, S) and (S, U, R) are complementary if T, F) and X, Y) are torsion theories in R-Mod and S-Mod, respectively. We find necessary and sufficient conditions on the bimodules RVS and SUR to have the complementarity of (R, V, S) and (S, U, R).

Skew group rings which are azumaya

If S is a ring with 1, and G is a finite group acting faithfully as automorphisms of S, then it is well known that the skew group ring S ∗ G is a separable extension of S if and only if there exists a central element in S with trace one. A ring is an Azumaya algebra if it is separable over its center. The case of when the group ring S[G] is Azumaya was studied by De Meyer and Janusz in [2] ; and lately the case of twisted group rings was studied by Szeto and Wong in [5] ; but the techniques used on those cases cannot be applied to the skew group rings precisely because the elements of the ring S do not commute with the elements of the group G. The purpose of this paper is to give conditions on the action of the group in order to make the skew group ring an Azumaya algebra. A ring A is said to be separable over a subring B if the (A − A)-module homomorphism of A⊗BA onto A defined by a⊗b 7−→ ab splits, and A is called H-separable over B if A ⊗B A is isomorphic as (A − A)-bimodule to a direct summand of A for some n ≥ 1. Clearly H-separable extension are separable. Let S be a ring with 1 and G be a finite group acting as automorphisms of S with fixed ring S; that we will denote R. The skew group ring S ∗G is the free left S-module with basis G, where multiplication is defined according

Dual dimension of modules over normalizing extensions

Let S = Si=1n Rai be a finite normalizing extension of R and suppose that SM is a left S-module. Denote by crk(A) the dual Goldie dimension of the module A. We show that crk(RM) = n · crk(SM) if either SM is artinian or the group homomorphism M ? aiM given by x ? aix is an isomorphism.

Exchange rings and decompositions of modules

In this paper, a new class of rings, called exchange rings, is defined and studied. The class of exchange rings includes regular rings in the sense of yon Neumann, local rings, and semiperfect rings. The chief property of these rings is that if S is any ring and M a left S-module which is a direct sum of finitely generated modules, each of which has an exchange ring as its endomorphism ring, then any two direct sum decompositions of M have isomorphic refinements. In particular, if R is an exchange ring, a projective R-module is a direct sum of ideals generated by idempotents. The main results of this paper are Theorems 2 and 3. Theorem 2 shows that a module has the finite exchange property (defined below) if and only if its endomorphism ring is an exchange ring. Theorem 3 gives a large class of rings which are exchange rings. The applications of Theorem 3 include generalizations of theorems proved by Kaplansky and Pierce for modules over regular rings. All rings will be associative rings with identity and all modules, unless otherwise specified, will be left modules. We will write the direct

On Morita Duality

A contra variant category-equivalence between categories of right R-modules and left S-modules (all rings have units, all modules are unitary) that contain RR, SS and are closed under submodules and factor modules, is naturally equivalent to a functor Horn (–, U) with a bimodule SUR such that SU, UR are injective cogenerators with S = End UR and R = End SU, and all modules in are [U-reflexive. Conversely, for any SSUR, Hom(–, U) is a contravariant category equivalence between the categories of [U-reflexive modules, and if U has the properties just stated, then these categories are closed under submodules, factor modules, and finite direct sums and contain RR, UR,SS, and SU. Such a functor will be called a (Morita) duality between R and S induced by U (see (5)).

Ring Theoretic Properties of Matrix Rings

K. Morita has shown that, given two rings R and S, there is an isomorphism between the category of left R-modules and the category of left S-modules if and only if there exists an R-S bimodule U such that(1) U is a progenerator in the category of left R-modules, and(2) S ≅ (EndR U)opp as rings.