Weakly automorphism invariant modules and essential tightness

While a module is pseudo-injective if and only if it is automorphism-invariant, it was not known whether automorphism-invariant modules are tight. It is shown that weakly automorphism-invariant modules are precisely essentially tight. We give various examples of weakly automorphism-invariant and essentially tight modules and study their properties. Some particular results: (1) R is a semiprime right and left Goldie ring if and only if every right (left) ideal is weakly injective if and only if every right (left) ideal is weakly automorphism invariant; (2) R is a CEP-ring if and only if R is right artinian and every indecomposable projective right R-module is uniform and essentially R-tight.     

On weakly prime radical of modules and semi-compatible modules

Summary Let M be a left R-module. Then a proper submodule P of M is called weakly prime submodule if for any ideals A and B of R and any submodule N of M such that ABN ⊆ P, we have AN ⊆ P or BN ⊆ P. We define weakly prime radicals of modules and show that for Ore domains, the study of weakly prime radicals of general modules reduces to that of torsion modules. We determine the weakly prime radical of any module over a commutative domain R with dim (R) ≦ 1. Also, we show that over a commutative domain R with dim (R) ≦ 1, every semiprime submodule of any module is an intersection of weakly prime submodules. Localization of a module over a commutative ring preserves the weakly prime property. An R-module M is called semi-compatible if every weakly prime submodule of M is an intersection of prime submodules. Also, a ring R is called semi-compatible if every R-module is semi-compatible. It is shown that any projective module over a commutative ring is semi-compatible and that a commutative Noetherian ring R is semi-compatible if and only if for every prime ideal B of R, the ring R/\B is a Dedekind domain. Finally, we show that if R is a UFD such that the free R-module R⊕ R is a semi-compatible module, then R is a Bezout domain.     

On the inheritance of the strongly π-extending property

In this article, we focus on modules with the property that every projection invariant submodule is essential in a fully invariant direct summand. In contrast to π-extending condition, it is shown that the former property is inherited by direct summands and Morita invariant. An application of our results yields that the endomorphism ring of a free module enjoys the property. Moreover, we characterize generalized triangular matrix rings with the aforementioned property and apply to somewhat special cases.     

Generating dual Baer modules via fully invariant submodules

Abstract

In this paper, we introduce a concept of a dual F-Baer module M where F is the fully invariant submodule of M, by this means we deal with generating dual Baer modules. We investigate direct sums of dual F -Baer modules M by exerting the notion of relatively dual F-Baer modules. We also obtain applications of dual F-Baer modules to rings and the preradical Z*(·).     

Weakly regular modules over normal rings

Under study are some conditions for the weakly regular modules to be closed under direct sums and the rings over which all modules are weakly regular. For an arbitrary right R-module M, we prove that every module in the category σ(M) is weakly regular if and only if each module in σ(M) is either semisimple or contains a nonzero M-injective submodule. We describe the normal rings over which all modules are weakly regular.     

On M-coidempotent elements and fully coidempotent modules

Abstract

In this article, we introduce the notion of M-coidempotent elements of a ring and investigate their connections with fully coidempotent modules, fully copure modules and vn-regular modules where M is a module. We prove that if M is a finitely cogenerated module, then M is fully copure if and only if M is semisimple. We prove that if M is a Noetherian module or M is a finitely cogenerated module, then M is fully coidempotent if and only if M is a vn-regular module. Finally, we give a characterization of semisimple Artinian modules via weak idempotents.     

Monomorphisms in the category of firm modules

Abstract

In this article, we consider the category of unitary right modules over an (associative) ring and the category of firm right modules over an idempotent ring. We study monomorphisms in these categories and give conditions under which morphisms are monomorphisms in the category of firm modules. We also prove that the lattice of categorically defined subobjects of a firm module is isomorphic to the lattice of unitary submodules of that module.     

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.     

McCoy modules and related modules over commutative rings

Let M be a left R-module. Then M is a McCoy (resp., dual McCoy) module if for nonzero f(X)∈R[X] and m(X)∈M[X], f(X)m(X) = 0 implies there exists a nonzero r∈R (resp., m∈M) with rm(X) = 0 (resp., f(X)m = 0). We show that for R commutative every R-module is dual McCoy, but give an example of a non-McCoy module. A number of other results concerning (dual) McCoy modules as well as arithmetical, Gaussian, and Armendariz modules are given.     

Graded Modules Over the q-Analog Virasoro-Like Algebra

In this paper, we deal with the classification of the irreducible Z-graded and Z ²-graded modules with finite dimensional homogeneous subspaces for the q analog Virasoro-like algebra L. We first prove that a Z-graded L-module must be a uniformly bounded module or a generalized highest weight module. Then we show that an irreducible generalized highest weight Z-graded module with finite dimensional homogeneous subspaces must be a highest (or lowest) weight module and give a necessary and sufficient condition for such a module with finite dimensional homogeneous subspaces. We use the Z-graded modules to construct a class of Z ²-graded irreducible generalized highest weight modules with finite dimensional homogeneous subspaces. Finally, we classify the Z ²-graded L-modules. We first prove that a Z ²-graded module must be either a uniformly bounded module or a generalized highest weight module. Then we prove that an irreducible nontrivial Z ²-graded module with finite dimensional homogeneous subspaces must be isomorphic to a module constructed as above. As a consequence, we also classify the irreducible Z-graded modules and the irreducible Z ²-graded modules with finite dimensional homogeneous subspaces and center acting nontrivial. Supported by the National Science Foundation of China (No 10671160), the China Postdoctoral Science Foundation (No. 20060390693), the Specialized Research fund for the Doctoral Program of Higher Education (No.20060384002), and the New Century Talents Supported Program from the Education Department of Fujian Province.     

Transfer of splitness with respect to a fully invariant short exact sequence in abelian categories

Abstract

We study the transfer via functors between abelian categories of the (dual) relative splitness of objects with respect to a fully invariant short exact sequence. We mainly consider fully faithful functors and adjoint pairs of functors. We deduce applications to Grothendieck categories, (graded) module categories and comodule categories.     

On dual of square free modules

A module M is said to be square free if whenever its submodule is isomorphic to N² = N⊕N for some module N, then N = 0. Dually, a module M is said to be d-square free (dual square free) if whenever its factor module is isomorphic to N² for some module N, then N = 0. In this paper, we give some fundamental properties of d-square free modules and study rings whose d-square free modules are closed under submodules or essential extensions.     

Modules having Baer summands

Let R be an arbitrary ring with identity and M a right R-module with S = End_R(M). Let F be a fully invariant submodule of M and I^?1(F) denotes the set {m∈M:Im?F} for any subset I of S. The module M is called F-Baer if I^?1(F) is a direct summand of M for every left ideal I of S. This work is devoted to the investigation of properties of F-Baer modules. We use F-Baer modules to decompose a module into two parts consists of a Baer module and a module determined by fully invariant submodule F, namely, for a module M, we show that M is F-Baer if and only if M = F⊕N where N is a Baer module. By using F-Baer modules, we obtain some new results for Baer rings.     

Another type of dimensions of modules and rings

Abstract

We say that a class Q of left R-modules is a monic class if a nonzero submodule of a module in Q is also a module in Q. For a monic class Q, we define a Q-dimension of modules that measures how far modules are from the modules in Q. For a monic class Q of indecomposable modules we characterize rings whose modules have Q-dimension. We prove that for an artinian principal ideal ring the Q-dimension coincides with the uniserial dimension. We also characterize when every module has Q-dimension.     

Weakly power automorphisms of groups

An automorphism α of a group G is called a weakly power automorphism if it maps every non-periodic subgroup of G onto itself. The aim of this paper is to investigate the behavior of weakly power automorphisms. In particular, among other results, it is proved that all weakly power automorphisms of a soluble non-periodic group G of derived length at most 3 are power automorphisms, i.e. they fix all subgroups of G. This result is best possible, as there exists a soluble non-periodic group of derived length 4 admitting a weakly power automorphism, which is not a power automorphism.     

Symmetric Group Modules with Specht and Dual Specht Filtrations

The author and Nakano recently proved that multiplicities in a Specht filtration of a symmetric group module are well-defined precisely when the characteristic is at least five. This result suggested the possibility of a symmetric group theory analogous to that of good filtrations and tilting modules for GL _n (k). This article is an initial attempt at such a theory. We obtain two sufficient conditions that ensure a module has a Specht filtration, and a formula for the filtration multiplicities. We then study the categories of modules that satisfy the conditions, in the process obtaining a new result on Specht module cohomology.

Next we consider symmetric group modules that have both Specht and dual Specht filtrations. Unlike tilting modules for GL _n (k), these modules need not be self-dual, and there is no nice tensor product theorem. We prove a correspondence between indecomposable self-dual modules with Specht filtrations and a collection of GL _n (k)-modules which behave like tilting modules under the tilting functor. We give some evidence that indecomposable self-dual symmetric group modules with Specht filtrations may be indecomposable self dual trivial source modules.     

Finitistic n-Self-Cotilting Modules

We study a class of modules which can be characterized using a duality theorem, called finitistic n-self-cotilting. Such a module Q can be characterized using dual conditions of some generalizations for star modules: every module M which has a right resolution with n terms isomorphic to finite powers of Q (i.e., M is n-finitely Q-copresented) has a right resolution with (n + 1) terms, and the functor Hom _R (?, Q) preserves the exactness of all monomorphisms with their ranges finite powers of Q and cokernels n-finitely Q-copresented modules. In the general case, these modules are independent toward other kinds of modules which are characterized using some dualities (w-Π _f -quasi injective modules, costar modules, f-cotilting modules). Closure properties for the classes involved in the duality are studied. At the end of the article, connections with the cotilting theory are exhibited, in the case of finitely dimensional algebras over fields.     

On Rings Having McCoy-Like Conditions

ABSTRACT

In this paper, we study the behavior of the couniform (or dual Goldie) dimension of a module under various polynomial extensions. For a ring automorphism σ ∈ Aut(R), we use the notion of a σ-compatible module M _R to obtain results on the couniform dimension of the polynomial modules M[x], M[x ^?1], and M[x, x ^?1] over suitable skew extension rings.