Advances in Ring Theory

Frontmatter

Kasch Modules

Abstract

An associative ring R is a left Kasch ring if it contains a copy of every simple left R-module. Transferring this notion to modules we call a left R-module М a Kasch module if it contains a copy of every simple module in σ [M]. The aim of this paper is to characterize and investigate this class of modules.

Toma Albu, Robert Wisbauer

Compactness in Categories and Interpretations

Abstract

It is the purpose of this note to give a definition for compactness of objects in a category in terms of a covariant functor F. Choosing the category and the functor F appropriately, we rediscover many kinds of compactnesses, so for instance, we get the usual compactness for T ₁-topological spaces and linear compactness of rings and modules. Generalizing a module-theoretic theorem of Leptin [6] we prove that if an object X of a locally small complete abelian category endowed with a linearly compact topology, has a dense semisimple subobject then X is a product of simple objects. We prove also a decomposition theorem for linearly compact rings into the product of simple rings, in particular, for linearly compact Brown-McCoy semisimple rings. Also a connection between compactness and sheaves is shown.

P. N. Ánh, R. Wiegandt

A Ring of Morita Context in Which Each Right Ideal is Weakly Self-Injective

Abstract

In this paper, among others, an example of a noetherian ring of Morita Context in which each right ideal is weakly self-injective, has been studied.

S. Barthwal, S. K. Jain, S. Jhingan, Sergio R. López-Permouth

Splitting Theorems and a Problem of Müller

Abstract

In this paper we introduce and investigate a condition (FI) which encompases a large class of rings including duo rings, FPF rings, and GFC rings. This condition is used in our main results to generalize a splitting theoren of C. Faith, and it is also used to provide a large class of self-injective rings on which a question of B. J. Müller has an affirmative answer.

Gary F. Birkenmeier, Jin Yong Kim, Jae Keol Park

Decompositions of D1 Modules

Abstract

Continuity and quasicontinuity for modules may be viewed as generalizations of quasi-injectivity. A key property of quasi-continuous modules is that complements are summands. Modules with this special property are called extending modules or C1 modules. We investigate decomposition properties of dual-extending (D1) modules, those modules which are supplemented and for which each supplement is a summand. The notions of hollowness and dual Goldie dimension play a prominent role. Our results are analogous to results for extending modules developed by Camillo and Yousif.

Robert A. Brown, Mary H. Wright

Right Cones in Groups

Abstract

A right cone C in a group G is a submonoid of G that generates G and aC ⊆ bC or bC ⊂ aC holds for any a, b in C; such a right cone is closely related to the cones of (right) linearly ordered groups on the one hand and valuation rings, in particular right chain domains, on the other. The ideal theory of right cones is described, the rank one right cones are classified, and three problems are raised.

H. H. Brungs, G. Törner

On Extensions of Regular Rings of Finite Index by Central Elements

Abstract

A Von Neumann regular ring R of finite index of nilpotency which is not biregular can have quite complex structure. Even though R can be embedded in a biregular ring of the same index, it need not be “close” to one in structure. It is, however, closely related to a unique smallest overring, R^#, which is “almost biregular”, i.e., one where supports of elements in the Pierce sheaf are open. It is formed by adjoining certain central idempotents from Q(R). Extensions of R by central elements, particularly idempotents, are examined. Many examples and counterexamples are presented.

W. D. Burgess, R. M. Raphael

Intersections of Modules

Abstract

Necessary and sufficient condition are found on a ring R or on a module M so that every submodule K < M can be represented as an irredundant intersection K = ⋂{K _i | i ∈ I} of submodules K < K _i < M, where either the K _i or their quotients M/K _i are required to satisfy some prescribed property. The phenomenon of irredundancy of such intersections is investigated.

John Dauns

Minimal Cogenerators Over Osofsky and Camillo Rings

Abstract

The direct sum C of the injective hulls E(V _i ) of the set {V _i } _{i ∈ I} of non-isomorphic simple right R-modules is a minimal right co-generator for R. While the injective hull E(C) is the unique (up to isomorphism) minimal injective right cogenerator, Osofsky [0] showed C is not necessarily unique even for commutative R, but that it is when R is either right Noetherian, semilocal, or C is quasiinjective. In this paper, we call a ring R a right Osofsky ring when C is the unique minimal right cogenerator, and show that rings studied by Camillo [Cl] with the property that Hom _R (E(V _i ), E(V _j )) = 0 for i ≠ j, are right Osofsky. We call these right Camillo rings, and show that commutative SISI rings of Vámos [V], and locally perfect commutative rings, in fact, any 0-dimensional ring, among others, are Camillo, hence Osofsky rings.

Carl Faith

Uniform Modules Over Goldie Prime Serial Rings

Abstract

We investigate the uniseriality of uniform injective modules over serial rings. Let R be an arbitrary ring and fix a decomposition, of the identity, 1 = e ₁ + e ₂ + ⋯ + e _n into orthogonal idempotents. For any uniform injective module V _R , we prove that there exists e = e _i such that, with A = eRe, V _R ≅ hom _A (Re,Ve). Moreover, Ve is a uniform injective A-module. We also show that if R is Goldie prime serial, then V is uniserial if and only if Ve is uniserial as an A-module.

Franco Guerriero

Co— Versus Contravariant Finiteness of Categories of Representations

Abstract

This article supplements recent work of the authors. (1) A criterion for failure of covariant finiteness of a full subcategory of Λ-mod is given, where Λ is a finite dimensional algebra. The criterion is applied to the category P ^∞ (Λ-mod) of all finitely generated Λ-modules of finite projective dimension, yielding a negative answer to the question whether P ^∞ (Λ-mod) is always covariantly finite in Λ-mod. Part (2) concerns contravariant finiteness of P ^∞ (Λ-mod). An example is given where this condition fails, the failure being, however, curable via a sequence of one-point extensions. In particular, this example demonstrates that curing failure of contravariant finiteness of P ^∞ (Λ-mod) usually involves a tradeoff with respect to other desirable qualities of the algebra.

B. Huisgen-Zimmermann, S. O. Smalø

Monomials and the Lexicographic Order

Abstract

This paper investigates the relationship between the lexicographic order on monomials and lex-segments of monomials

Heather Hulett

Rings Over Which Direct Sums of CS Modules are CS

Abstract

A module M is defined to be a CS module, if every submodule of M is essential in a direct summand of M. In this paper we show, among other results, that for a right nonsingular ring R, all direct sums of CS right R-modules are CS if and only if R is a right artinian ring and every indecomposable infective right R-module has length at most two.

Dinh Van Huynh, Bruno J. Müller

Exchange Properties and the Total

Abstract

We study several exchange properties for modules for their behavior under the formation of direct sums and their relationship with the total of a module.

Friedrich Kasch, Wolfgang Schneider

Local Bijective Gabriel Correspondence and Torsion Theoretic FBN Rings

Abstract

A right noetherian ring R that has local bijective Gabriel correspondence with respect to a torsion theory τ need in general not be right fully τ-bounded, but it is, if and only if the τ -closed prime ideals satisfy a version of the second layer condition. Rings with local bijective Gabriel correspondence are characterized by the tameness of their τ -torsionfree modules, and this implies that their relative (Gabriel-Rentschler) Krull and classical Krull dimensions coincide.

Paul Kim, Günter Krause

Normalizing Extensions and the Second Layer Condition

Abstract

We characterize the second layer condition for a link closed subset of Spec(S) where S is a Noetherian normalizing extension of a Noetherian ring R and R satisfies the second layer condition. The second layer condition is shown to depend on the R-module structure of tame injective S-modules that are naturally associated with prime ideals in the link closed set. This is used to demonstrate that certain twisted polynomial rings satisfy the second layer condition when R is the coefficient ring. In case S is a centralized extension, our characterization is applied to show that the strong second layer condition for S amounts to a diluted version of AR-separation for S whenever R is AR-separated.

Karl A. Kosler

Generators of Subgroups of Finite Index in GL m (ℤG)

Abstract

Let G be a finite group, and ℤG its integral group ring. We provide a set of generators of a subgroup of finite index in the general linear group, GL _m (ℤG), provided m ≥ 3. We also provide partial results in the case m = 2.

Gregory T. Lee, Sudarshan K. Sehgal

Weak Relative Injective M-Subgenerated Modules

Abstract

We study weak relative injective and relative tight modules in the category σ[M], where М is a right R-module. Many of the known results in the category of right R-modules are extended to σ[М]without assuming either М is projective or finitely generated. Conditions are given for a A-tight module to be weakly A-injective in σ[M]. Modules for which every submodule is weakly injective (tight) in σ[М] are characterized. Modules М for which every module in σ[М] is weakly injective and for which weakly injective modules are closed under direct sums are studied.

Saroj Malik, N. Vanaja

Direct Product and Power Series Formations Over 2-Primal Rings

Abstract

We show that the direct product of an infinite set of 2-primal rings (or even rings satisfying (PS I)) need not be a 2-primal ring, and we develop some sufficient conditions on the rings for their direct product to be 2-primal. We also show that the ring of formal power series over a 2-primal ring (or even a ring satisfying (PS I)) need not be 2-primal.

Greg Marks

Localization in Noetherian Rings

Abstract

When A is a left Noetherian ring with nilradical N, then there is a unitary subring B of A and ∑ a left denominator set in B such that Q, the ring of left fractions of B with respect to ∑ is left Artinian. Furthermore, for P = Q ⊗ _B A, P is a flat right A-module of type FP such that M, a left A-module, is C(N)-torsion if and only if P ⊗ _A M = 0. For the functors T = P ⊗ _A (·): A mоd→ Q-mod and S = Hom _Q (P,·): Q-mod→ A-mod, the natural transformation 1 → ST, M ↦ ST(M) is the localization of M in A-mod with respect to the torsion theory on A-mod corresponding to the multiplicative set C(N).

When I is a semiprime ideal of a left Noetherian ring, then for each positive integer n, a ring Q _n is constructed as above for N = I/I ⁿ the nilradical of A/ I ⁿ and a sequence Q _{n +1} → Q _n of surjective ring homomorphisms with inverse limit Q a semiperfect ring.

Michael Mcconnell, Francis L. Sandomierski

Projective Dimension of Ideals in von Neumann Regular Rings

Abstract

This paper is motivated by an attempt to solve an old problem of Wiegand, which asks whether the projective dimension of an ideal in a commutative von Neumann regular ring depends only on the lattice of idempotents in that ideal. We compute the projective dimension of some infinitely generated ideals in von Neumann regular rings. In previous work, this projective dimension, if computable, was either ‘obvious’ or the subscript of the aleph of a generating set. We give nontrivial examples which can have arbitrary preassigned projective dimension and arbitrarily large cardinality of a generating set. The paper then presents a function ℓ from the class of all nonzero submodules of projective modules over a von Neumann regular ring to the class of all ordinals. This function depends only on the lattice of cyclic submodules of M. We show that ℓ (М) = 0 ⇔ M is projective and ℓ (M) ≥ pd (M). We conjecture that pd (M) < ∞ ⇒ pd (M) = ℓ (M) for all M. Since ℓ (M) is defined lattice theoretically, this would answer Wiegand’s question affirmatively. Even if our conjecture is false, ℓ (М) seems like an interesting lattice invariant to explore.

Barbara L. Osofsky

Homological Properties of Color Lie Superalgebras

Abstract

Let L = L ₊⊕ L ₋ be a finite dimensional color Lie superalgebra over a field of characteristic 0 with universal enveloping algebra U(L). We show that gldim(U(L ₊)) = IFPD(U(L) = rFPD(U(L)) = injdim _U(L) (U(L)) = dim(L₊). We also prove that U(L) is Auslander-Gorenstein and Cohen-Macaulay and thus that it has a QF classical quotient ring.

Kenneth L. Price

Indecomposable Modules Over Artinian Right Serial Rings

Abstract

Let R be an artinian ring with Jacobson radical J such that J ² = 0 and R/J is a direct product of matrix rings over finite dimensional division rings. The structure of R is determined, in case every indecomposable right R-module is uniform. Furthermore, all indecomposable right or left modules over such a ring are determined.

Surjeet Singh

Nonsingular Extending Modules

Abstract

In this paper, it is shown that if R is a semiprime right Goldie ring, then any nonsingular extending right R-module is the direct sum of an injective module and a finite number of uniform modules.

Patrick F. Smith

Right Hereditary, Right Perfect Rings are Semiprimary

Abstract

If R is a right hereditary, right perfect ring, then every maximal ideal of R is idempotent, and every ideal of R has a stationary power. Consequently, every right hereditary, right perfect ring must be semiprimary.

Mark L. Teply

On the Endomorphism Ring of a Discrete Module: A Theorem of F. Kasch

Abstract

Using results from the monograph of Mohamed and Mueller on discrete modules, it is shown that the endomorphism ring of a discrete module, modulo its radical, is a direct product of full linear rings.

Julius M. Zelmanowitz

Nonsingular Rings with Finite Type Dimension

Abstract

Two modules are said to be orthogonal if they do not have nonzero isomorphic submodules. An atomic module is any nonzero module whose nonzero submodules are not orthogonal. A module is said to have type dimension n if it contains an essential submodule which is a direct sum of n pairwise orthogonal atomic submodules; If such a number n does not exist, we say the type dimension of this module is ∞. In this paper, we provide characterizations and examples of nonsingular rings with finite type dimension. A characterization theorem is proved for nonsingular rings whose nonzero right ideals contain nonzero atomic right ideals. Type dimension formulas are also obtained for polynomial rings, Laurent polynomial rings and formal triangular matrix rings.

Yiqiang Zhou

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