Rickart Modules

The concept of right Rickart rings (or right p.p. rings) has been extensively studied in the literature. In this article, we study the notion of Rickart modules in the general module theoretic setting by utilizing the endomorphism ring of a module. We provide several characterizations of Rickart modules and study their properties. It is shown that the class of rings R for which every right R-module is Rickart is precisely that of semisimple artinian rings, while the class of rings R for which every free R-module is Rickart is precisely that of right hereditary rings. Connections between a Rickart module and its endomorphism ring are studied. A characterization of precisely when the endomorphism ring of a Rickart module will be a right Rickart ring is provided. We prove that a Rickart module with no infinite set of nonzero orthogonal idempotents in its endomorphism ring is precisely a Baer module. We show that a finitely generated module over a principal ideal domain (PID) is Rickart exactly if it is either semisimple or torsion-free. Examples which delineate the concepts and results are provided.     

Essential extensions, excellent extensions and finitely presented dimension

In Section 1 of this paper, we investigate the finitely presented dimension of an essential extension for a module, and obtain results concerning an essential extension of a torsion-free module. We partially answer the question: When is an essential extension of a finitely presented module (an almost finitely presented module) also finitely presented (almost finitely presented)? In Section 2, we study theC-excellent extensions and the finitely presented dimensions. We obtain some results on the homological dimensions of matrix rings and group rings.     

Class and rank of differential modules

A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class – a substitute for the length of a free complex – and on the rank of a differential module in terms of invariants of its homology. These results specialize to basic theorems in commutative algebra and algebraic topology. One instance is a common generalization of the equicharacteristic case of the New Intersection Theorem of Hochster, Peskine, P. Roberts, and Szpiro, concerning complexes over commutative noetherian rings, and of a theorem of G. Carlsson on differential graded modules over graded polynomial rings.     

Local rings whose modules are almost serial

The present paper is a sequel to our previous work on almost uniserial rings and modules, which appeared in the Journal of Algebra in 2016; it studies rings over which every (left and right) module is almost serial. A module is almost uniserial if any two of its submodules are either comparable in inclusion or isomorphic. And a module is almost serial if it is a direct sum of almost uniserial modules. The results of the paper are inspired by a characterization of Artinian serial rings as rings having all left (or right) modules serial. We prove that if R is a local ring and all left R-modules are almost serial then R is an Artinian ring which is uniserial either on the left or on the right. We also produce a connection between local rings having all left and right modules almost serial, local balanced rings studied by Dlab and Ringel and local Köthe rings. Finally we prove Morita invariance of the almost serial property and list some consequences.     

The topological jacobson radical of rings. II

In this paper, topologically primitive rings and rings possessing a faithful topologically irreducible module and bounded by this module are considered for the investigation of properties of their topological Jacobson radical. We investigate the topological Jacobson radical in some classes of topological rings such as left topologically Artinian rings, topological rings possessing a basis of neighborhoods of zero consisting of ideals, compact rings, and bounded strictly linearly compact rings.     

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.     

Gorenstein Projective Precovers

We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In Sect. 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Gorenstein projective dimension) and that are not right coherent.     

Regular semiartinian rings

We study the structure of rings over which every right module is an essential extension of a semisimple module by an injective one. A ring R is called a right max-ring if every nonzero right R-module has a maximal submodule. We describe normal regular semiartinian rings whose endomorphism ring of the minimal injective cogenerator is a max-ring.     

The Rings Characterized by Minimal Left Ideals

We study these rings with every minimal left ideal being a projective, direct summand and a p-injective module, respectively. Some characterizations of these rings are given, and the relations among them are obtained. With these rings, we characterize seinisiinple rings. Finally, we introduce MC2 rings, and give some characterizations of MC2 rings.     

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.     

Strongly compressible modules and semiprime right goldie rings

A module M _R is defined to be strongly compressible (or SC for short) if for every essential submodule N of M, there exists X ? E(M) such that M ? X ? N. We show that a ring R is semiprime right Goldie iff R _Ris SC module iff every right ideal of R is SC module iff every submodule of each progenerator of Mod-R is SC module. As corollaries of this result, we obtain some new module-theoretic characterizations of semiprime Goldie rings, prime (right) Goldie rings and Prüfer rings, etc., etc.,respectively. And the characterization theorem of semiprime Goldie rings of López-Permouth, Rizvi and Yousif by using weakly-injective modules can be regarded as a corollary of our results.     

On the Left Perpendicular Category of the Modules of Finite Projective Dimension

In this article, we characterize several properties of commutative noetherian local rings in terms of the left perpendicular category of the category of finitely generated modules of finite projective dimension. As an application, we prove that a local ring is regular if (and only if) there exists a strong test module for projectivity having finite projective dimension. We also obtain corresponding results with respect to a semidualizing module.     

GORENSTEIN MODULES,FINITE INDEX,AND FINITE COHEN–MACAULAY TYPE

ABSTRACT

A Gorenstein module over a local ring R is a maximal Cohen–Macaulay module of finite injective dimension. We use existence of Gorenstein modules to extend a result due to S. Ding: A Cohen–Macaulay ring of finite index, with a Gorenstein module, is Gorenstein on the punctured spectrum. We use this to show that a Cohen–Macaulay local ring of finite Cohen–Macaulay type is Gorenstein on the punctured spectrum. Finally, we show that for a large class of rings (including all excellent rings), the Gorenstein locus of a finitely generated module is an open set in the Zariski topology.     

Cohen–Macaulay Modules and Holonomic Modules Over Filtered Rings

We study Gorenstein dimension and grade of a module M over a filtered ring whose associated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G?dim M ≤ G?dim gr M and an equality grade M = grade gr M, whenever Gorenstein dimension of gr M is finite (Theorems 2.3 and 2.8). We would say that the use of G-dimension adds a new viewpoint for studying filtered rings and modules. We apply these results to a filtered ring with a Cohen–Macaulay or Gorenstein associated graded ring and study a Cohen–Macaulay, perfect, or holonomic module.     

Computations of atom spectra

This is a contribution to the theory of atoms in abelian categories recently developed in a series of papers by Kanda. We present a method that enables one to explicitly compute the atom spectrum of the module category over a wide range of non‐commutative rings. We illustrate our method and results by several examples.     

Modules in Which Every Fully Invariant Submodule is Essential in a Direct Summand

A module M is called extending if every submodule of M is essential in a direct summand. We call a module FI-extending if every fully invariant submodule is essential in a direct summand. Initially we develop basic properties in the general module setting. For example, in contrast to extending modules, a direct sum of FI-extending modules is FI-extending. Later we largely focus on the specific case when a ring is FI-extending (considered as a module over itself). Again, unlike the extending property, the FI-extending property is shown to carry over to matrix rings. Several results on ring direct decompositions of FI-extending rings are obtained, including a proper generalization of a result of C. Faith on the splitting-off of the maximal regular ideal in a continuous ring.     

Global Weyl modules for the twisted loop algebra

We define global Weyl modules for twisted loop algebras and analyze their highest weight spaces, which are in fact isomorphic to Laurent polynomial rings in finitely many variables. We are able to show that the global Weyl module is a free module of finite rank over these rings. Furthermore we prove, that there exist injective maps from the global Weyl modules for twisted loop algebras into a direct sum of global Weyl modules for untwisted loop algebras. Relations between local Weyl modules for twisted and untwisted generalized current algebras are known; we provide for the first time a relation on global Weyl modules.