Generalized <Emphasis Type="Italic">SV</Emphasis>-modules

Given an arbitrary quasiprojective right R-module P, we prove that every module in the category σ(P) is weakly regular if and only if every module in σ(M/I(M)) is lifting, where M is a generating object in σ(P). In particular, we describe the rings over which every right module is weakly regular.     

On rings with associated elements

A principal right ideal of a ring is called uniquely generated if any two elements of the ring that generate the same principal right ideal must be right associated (i.e., if for all a,b in a ring R, aR = bR implies a = bu for some unit u of R). In the present paper, we study “uniquely generated modules” as a module theoretic version of “uniquely generated ideals,” and we obtain a characterization of a unit-regular endomorphism ring of a module in terms of certain uniquely generated submodules of the module among some other results: End(M) is unit-regular if and only if End(M) is regular and all M-cyclic submodules of a right R-module M are uniquely generated. We also consider the questions of when an arbitrary element of a ring is associated to an element with a certain property. For example, we consider this question for the ring R[x;σ]∕(xⁿ⁺¹), where R is a strongly regular ring with an endomorphism σ be an endomorphism of R.     

Strongly Duo Modules and Rings

An R-module M is called strongly duo if Tr(N, M) = N for every N ≤ M _R . Several equivalent conditions to being strongly duo are given. If M _R is strongly duo and reduced, then End _R (M) is a strongly regular ring and the converse is true when R is a Dedekind domain and M _R is torsion. Over certain rings, nonsingular strongly duo modules are precisely regular duo modules. If R is a Dedekind domain, then M _R is strongly duo if and only if either M ≈ R or M _R is torsion and duo. Over a commutative ring, strongly duo modules are precisely pq-injective duo modules and every projective strongly duo module is a multiplication module. A ring R is called right strongly duo if R _R is strongly duo. Strongly regular rings are precisely reduced (right) strongly duo rings. A ring R is Noetherian and all of its factor rings are right strongly duo if and only if R is a serial Artinian right duo ring.     

Weakly Koszul-like modules

The Koszul-like property for any finitely generated graded modules over a Koszul-like algebra is investigated and the notion of weakly Koszul-like module is introduced. We show that a finitely generated graded module M is a weakly Koszul-like module if and only if it can be approximated by Koszul-like graded submodules, which is equivalent to the fact that G(M) is a Koszul-like module, where G(M) denotes the associated graded module of M. As applications, the relationships between minimal graded projective resolutions of M and G(M), and Koszul-like submodules are established. Moreover, the Koszul dual of a weakly Koszul-like module is proved to be generated in degree 0 as a graded E(A)-module.     

On Fully Idempotent Modules

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.     

Modules over Endomorphism Rings

It is proved that A is a right distributive ring if and only if all quasiinjective right A-modules are Bezout left modules over their endomorphism rings if and only if for any quasiinjective right A-module M which is a Bezout left End (M)-module, every direct summand N of M is a Bezout left End(N)-module. If A is a right or left perfect ring, then all right A-modules are Bezout left modules over their endomorphism rings if and only if all right A-modules are distributive left modules over their endomorphism rings if and only if A is a distributive ring.     

Rugged modules: The opposite of flatness

Relative notions of flatness are introduced as a mean to gauge the extent of the flatness of any given module. Every module is thus endowed with a flatness domain and, for every ring, the collection of flatness domains of all of its modules is a lattice with respect to class inclusion. This lattice, the flatness profile of the ring, allows us, in particular, to focus on modules which have a smallest flatness domain (namely, one consisting of all regular modules.) We establish that such modules exist over arbitrary rings and we call them Rugged Modules. Rings all of whose (cyclic) modules are rugged are shown to be precisely the von Neumann regular rings. We consider rings without a flatness middle class (i.e., rings for which modules must be either flat or rugged.) We obtain that, over a right Noetherian ring every left module is rugged or flat if and only if every right module is poor or injective if and only if R = S×T, where S is semisimple Artinian and T is either Morita equivalent to a right PCI-domain, or T is right Artinian whose Jacobson radical properly contains no nonzero ideals. Character modules serve to bridge results about flatness and injectivity profiles; in particular, connections between rugged and poor modules are explored. If R is a ring whose regular left modules are semisimple, then a right module M is rugged if and only if its character left module M⁺ is poor. Rugged Abelian groups are fully characterized and shown to coincide precisely with injectively poor and projectively poor Abelian groups. Also, in order to get a feel for the class of rugged modules over an arbitrary ring, we consider the homological ubiquity of rugged modules in the category of all modules in terms of the feasibility of rugged precovers and covers for arbitrary modules.     

Quasi Co-Hopfian Modules and Applications

We carry out an extensive study of modules M _R with the property that M/f(M) is singular for all injective endomorphisms f of M. Such modules called “quasi co-Hopfian”, generalize co-Hopfian modules. It is shown that a ring R is semisimple if and only if every quasi co-Hopfian R-module is co-Hopfian. Every module contains a unique largest fully invariant quasi co-Hopfian submodule. This submodule is determined for some modules including the semisimple ones. Over right nonsingular rings several equivalent conditions to being quasi co-Hopfian are given. Modules with all submodules quasi co-Hopfian are called “completely quasi co-Hopfian” (cqcH). Over right nonsingular rings and over certain right Noetherian rings, it is proved that every finite reduced rank module is cqcH. For a right nonsingular ring which is right semi-Artinian (resp. right FBN) the class of cqcH modules is the same as the class of finite reduced rank modules if and only if there are only finitely many isomorphism classes of nonsingular R-modules which are simple (resp. indecomposable injective).     

Modules Whose Endomorphism Rings are Division Rings

The well-known Schur's Lemma states that the endomorphism ring of a simple module is a division ring. But the converse is not true in general. In this paper we study modules whose endomorphism rings are division rings. We first reduce our consideration to the case of faithful modules with this property. Using the existence of such modules, we obtain results on a new notion which generalizes that of primitive rings. When R is a full or triangular matrix ring over a commutative ring, a structure theorem is proved for an R-module M such that End _R (M) is a division ring. A number of examples are given to illustrate our results and to motivate further study on this topic.     

Minimalities and Modules over Dedekind-Like Rings

We are interested in (right) modules M satisfying the following weak divisibility condition: If R is the underlying ring, then for every r ∈ R either Mr = 0 or Mr = M. Over a commutative ring, this is equivalent to say that M is connected with regular generics. Over arbitrary rings, modules which are “minimal” in several model theoretic senses satisfy this condition. In this article, we investigate modules with this weak divisibility property over Dedekind-like rings and over other related classes of rings.     

T-Semisimple Modules and T-Semisimple Rings

We define and investigate t-semisimple modules as a generalization of semisimple modules. A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t-essential in N. T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules. Many equivalent conditions for a module M to be t-semisimple are found. Accordingly, M is t-semisiple, if and only if, M = Z ₂(M) ⊕ S(M) (where Z ₂(M) is the Goldie torsion submodule and S(M) is the sum of nonsingular simple submodules). A ring R is called right t-semisimple if R _R is t-semisimple. Various characterizations of right t-semisimple rings are given. For some types of rings, conditions equivalent to being t-semisimple are found, and this property is investigated in terms of chain conditions.     

Strongly FP-injective modules

A module M is called strongly FP-injective if Extⁱ(P,M) = 0 for any finitely presented module P and all i≥1. (Pre)envelopes and (pre)covers by strongly FP-injective modules are studied. We also use these modules to characterize coherent rings. An example is given to show that (strongly) FP-injective (pre)covers may fail to be exist in general. We also give an example of a module that is FP-injective but not strongly FP-injective.     

Lifting Modules with Indecomposable Decompositions

A module M is called a “lifting module” if, any submodule A of M contains a direct summand B of M such that A/B is small in M/B. This is a generalization of projective modules over perfect rings as well as the dual of extending modules. It is well known that an extending module with ascending chain condition (a.c.c.) on the annihilators of its elements is a direct sum of indecomposable modules. If and when a lifting module has such a decomposition is not known in general. In this article, among other results, we prove that a lifting module M is a direct sum of indecomposable modules if (i) rad(M ^(I)) is small in M ^(I) for every index set I, or, (ii) M has a.c.c. on the annihilators of (certain) elements, and rad(M) is small in M.     

Rings Whose Modules Have Grade Zero

In this paper, we prove that R is a two-sided Artinian ring and J is a right annihilator ideal if and only if (i) for any nonzero right module, there is a nonzero linear map from it to a projective module; (ii) every submodule of RR is not a radical module for some right coherent rings. We call a ring a right X ring if Homa(M, R) = 0 for any right module M implies that M = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover we characterize semisimple rings by using X rings. A famous Faith‘s conjecture is whether a semipimary PF ring is a QF ring. Similarly we study the relationship between X rings and QF and get many interesting results.     

Principally quasi-injective modules

An R-module M is called principally quasi-injective if each R-hornomorphism from a principal submodule of M to M can be extended to an endomorphism of M. Many properties of principally injective rings and quasi-injective modules are extended to these modules. As one application, we show that, for a finite-dimensional quasi-injective module M in which every maximal uniform submodule is fully invariant, there is a bijection between the set of indecomposable summands of M and the maximal left ideals of the endomorphism ring of M

Throughout this paper all rings R are associative with unity, and all modules are unital. We denote the Jacobson radical, the socle and the singular submodule of a module M by J(M), soc(M) and Z(M), respectively, and we write J(M) = J. The notation N ?^ess M means that N is an essential submodule of M.     

Regularity and Substructures of Hom

ABSTRACT

We define “regular” for maps in a Hom group. This notion specializes to the well-known notions of (Von Neumann) regular in rings and modules. A map f ∈ Hom _R (A,M) is regular if and only if Ker(f) ?^⊕ A and Im(f) ?^⊕ M. There exists a unique maximal regular End(M)-End(A)-submodule in Hom _R (A,M). We study regularity in Hom _R (A ₁ ⊕ A ₂, M ₁ ⊕ M ₂). The existence of a regular function Hom _R (A,M) implies the existence of projective summands of Hom _R (A,M) _{End R (A)} and of _{End R ( M )} Hom _R (A,M). We consider regularity in endomorphism rings, and generalize a theorem of Ware-Zelmanowitz. We examine connections between the maximum regular bimodule and other substructures of Hom, mention two generalizations of regularity, and raise some questions.     

The homomorphismM*⊗N → Hom (M,N)

In this note we prove two theorems. In theorem 1 we prove that if M andN are two non-zero reflexive modules of finite projective dimensions over a Gorenstein local ring, such that Hom (M, N) is a third module of syzygies, then the natural homomorphismM* ⊗N → Hom (M, N) is an isomorphism. This extends the result in [7]. In theorem 2, we prove that projective dimension of a moduleM over a regular local ringR is less than or equal ton if and only if Ext_R ⁿ (M, R) ⊗M → Ext_R ⁿ (M, M) is surjective; in which case it is actually bijective. This extends the usual criterion for the projectivity of a module.     

Classes of Commutative Rings Defined by Special Conditions

ABSTRACT

In this article, we are mainly concerned with (n, d)-Krull rings, i.e., rings in which each n-presented prime ideal has height at most d. Precisely, we show that weakly n-Von Neumann regular rings are (n ? 1, 0)-Krull rings. Also, we prove that (n, d)-Krull property is not local property and that R is an (n, d)-Krull ring if and only if dim(R _P ) ≤ d for each n-presented prime ideal P of R. Finally, we construct a class of (2, d)-Krull rings which are neither (2, d ? 1)-Krull rings (for d = 1) nor (1, d)-Krull rings for d = 0,1.