Quasi-projective covers and direct sums

Krull–Schmidt categories and projective covers

has characterized those rings with involution J, with 1, and such that every symmetric element has an inverse under the additional hypothesis that S~~, the subring generated by ,S (the set of symmetric elements), is A, the ring itself.In this paper we raise the question as to when S~ = A under a weaker hypothesis and prove the following theorem.(A henceforth is such that 2A =A and 2x=0 implies x=0 for any xEA.)Theorem 1.Let A be a ring with 1 and suppose that S is a simple Jordan ring under the Jordan multiplication, s o t = st+ts for all s and t in S. Then either S~ = A or SEZ, the center of A, or K, the set o/ skew elements, is an ideal o/ A with K2EZ, K3 = (8), and S is an associative ring under Jordan multiplication.We note that Osborn proves that under his hypothesis, either A is a division ring; the direct sum of two division rings which are antiisomorphic, and / interchanges the summands; or 51 is a field under Jordan multiplication, K is an ideal of A where K2 = (d).A corollary to our result is Corollary 1.Let A be as in Theorem 1. Suppose /urther that A contains no nilpotent ideals.Then either A is simple or A is a direct sum o/ two simple rings which are anti-isomorphic and such that J interchanges the direct summands.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

The lattice structure of preradicals III: operators

Let R be a hereditary noetherian prime ring ((hnp)-ring) with enough invertible ideals. Torsion modules over bounded (hnp)-rings were studied by the author in [10; 11]. All the results proved in [10; 11] also hold for torsion R-modules having no completely faithful submodules. In Section 2, indecomposable injective torsion R-modules which are not completely faithful are studied, and they are shown to have finite periodicities (Theorem (2.8) and Corollary (2.9)). These results are used to determine the structure of quasi-injective and quasi-projective modules over bounded (hnp)-rings (Theorems (2.13), (2.14) and (2.15)).

Associative rings R and R′ are said to be lattice-isomorphic if their subring lattices L(R) and L(R′) are isomorphic. An isomorphism of the lattice L(R) onto the lattice L(R′) is called a projection (or else a lattice isomorphism) of the ring R onto the ring R′. A ring R′ is called the projective image of a ring R. Lattice isomorphisms of finite one-generated rings with identity are studied. We elucidate the general structure of finite one-generated rings with identity and also give necessary and sufficient conditions for a finite ring decomposable into a direct sum of Galois rings to be generated by one element. Conditions are found under which the projective image of a ring decomposable into a direct sum of finite fields is a one-generated ring. We look at lattice isomorphisms of one-generated rings decomposable into direct sums of Galois rings of different types. Three main types of Galois rings are distinguished: finite fields, rings generated by idempotents, and rings of the form GR(pn,m), where m > 1 and n > 1. We specify sufficient conditions for the projective image of a one-generated ring decomposable into a sum of Galois rings and a nil ideal to be generated by one element.

Let R be an associative ring with identity.Then the left socle of R is a direct summand of R as a right ϋNmodule if and only if it is projective as a left jR-module and contains no infinite sets of orthogonal idempotents.This implies, for example, that a ring with finitely generated left socle and no nilpotent minimal left ideals is a ring direct sum of a semisimple artinian ring and a ring with zero left socle.

Let R be an associative ring with identity. We introduce the notion of semi-τ-supplemented modules, which is adapted from srs-modules, for a preradical τ on R-Mod. We provide basic properties of these modules. In particular, we study the objects of R-Mod for τ = Rad. We show that the class of semi-τ-supplemented modules is closed under finite sums and factor modules. We prove that, for an idempotent preradical τ on R-Mod, a module M is semi-τ-supplemented if and only if it is τ-supplemented. For τ = Rad, over a local ring every left module is semi-Rad-supplemented. We also prove that a commutative semilocal ring whose semi-Rad-supplemented modules are a direct sum of w-local left modules is an artinian principal ideal ring.

Continuing an earlier examination of the codominant dimension of rings and modules, a categorical characterization is given for rings of equivalent dominant and codominant dimensions. Specifically, the question is reduced to when R R and the left minimal injective cogenerator R can be used as test modules respectively for the dominant dimension of the projective modules and the codominant dimension of the injective modules. These conditions are in turn characterized by when the injective projective modules are Z-injective. Also, a new and shortened version is given for the proof that the codominant dimension of the injective modules is equal to the dominant dimension of the projective modules. Throughout, R is an associative ring with unity and all modules considered are unital. We denote a minimal injective cogenerator for R)R by R such being isomorphic to an injective hull of a direct sum of a complete representative class of simple left R-modules. For a left perfect ring R and a left R-module RM we define the codominant dimension of RM, denoted codom dim RM, to be greater than or equal to n if for a minimal projective resolution . P* P M 0, Pi is injective for i = 1, 29, . . ., n. The left codominant dimesnion of R is defined to be codom dim RU. The dominant dimension of RM is defined in a dual manner, and the left dominant dimension of R is dom dimRR. Further, for R left perfect, we define codom dimR9 to be infIcodom dim RE: RE injectivel. Similarly, dom dim Ris given by inf dom dimRP: RP projectivel. Finally, we say that RE (nRP) is a test module for RI (RT) provided codom dim RE= codom dimRI (dom dimR P = dom dimRY). In [2] we showed that dom dim R 9= codom dim RI. As an application it can Presented to the Society, January 25, 1973; received by the editors April 16, 1973. AMS (MOS) subject classifications (1970). Primary 16A60; Secondary 16A50, 16A52.

Introduction. A nil group is an (additively written) abelian group G such that every associative ring R with G as its additive group has trivial multiplication, i.e., xy = 0 for every x, y ER. Szele [2] has shown that every nil group is either a torsion group (every element of finite order) or a torsion-free group (every element except the identity of infinite order), and he has completely characterized the torsion nil groups. In [3 ], Szele discusses some questions related to nil groups, but is unable to find conditions for the existence of torsionfree nil groups. An additive group G is said to be strongly nil if there is no nontrivial ring, associative or nonassociative, which has G as its additive group. It is the purpose of this note to characterize those nil and strongly nil groups which are (isomorphic to) a weak direct sum of subgroups of the additive group of rationals, and to show that every such nil group is also strongly nil. The authors would like to express thanks to the referee for helpful suggestions.

Let R be an associative ring with identity and let M be a unitary left R-module. In this paper, we introduce a ⊕-Jacobson-supplemented module (for short ⊕-J-supplemented), as a generalization of J-supplemented module. We state the main properties of a ⊕-J-supplemented modules and supplying by examples and remarks for these module and several properties of this modules are given. Also we introduce weakly ⊕-J-supplemented modules and give a characterization of weakly ⊕-J-supplemented modules a conditions under which the direct sum of weakly ⊕-J-supplemented modules is weakly ⊕-J-supplemented are given. We define a cofinitely ⊕-J-supplemented modules (for short, cof-⊕-J-supplemented) and some type of modules that are related to cof-⊕-J-supplemented module and prove properties of this type of modules. Also we discuss the relation between them with examples and remarks.

Weakly injective and weakly projective modules, which are extensions of the classes of quasi-injective modules and quasi-projective modules respectively, are studied. Descriptions are obtained for weakly injective and weakly projective Abelian groups, as well as for some classes of Abelian groups closely related to them.

In this paper a number of theorems are obtained on decompositions of Artinian alternative rings, generalizing analogous theorems of Szász, Šneĭdmjuller, Kertész et al. for associative Artinian rings. For example, it is shown that every alternative Artinian ring decomposes into a direct sum of torsion parts and a torsion free ideal. With several additional restrictions, we obtain a decomposition of any alternative Artinian ring into a direct sum of an ideal and a left ideal, allowing a rather detailed description. A description of hereditarily Artinian alternative rings is given. Bibliography: 16 titles.

This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2). It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications. The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.