CS Modules Relative to a Torsion Theory

In this article we introduce two new concepts, those of a τ-CS and a s-τ-CS module, which are both torsion-theoretic analogues of CS modules. We investigate their relationship with the familiar concepts of τ-injective, τ-simple and τ-uniform modules and compare them with τ-complemented (τ-injective) modules, which were considered by other authors as torsion-theoretic analogues of CS modules. We are interested in decomposing a relatively CS module into indecomposable submodules, and in determining when a direct sum of relatively CS modules is relatively CS. This paper forms part of the Ph.D. thesis of the first author, written under the supervision of the second author. The first author gratefully acknowledges the support of the Commonwealth Scholarship and Fellowship Committee of New Zealand.     

A CONTRIBUTION OF ADAM SULIŃSKI TO RADICAL THEORY

Abstract

The aim of this paper is to present a contribution of Adam Suliński to radical theory.     

NOTES ON NILPOTENCY OF NIL SUBRINGS OF ENDOMORPHISM RINGS OF MODULES

Abstract

A family K of right R-modules is called a natural class if K is closed under submodules, direct sums, infective hulls, and isomorphic copies. The main result of this note is the following: Let K be a natural class on Mod-R and M ε K. If M satisfies a.c.c. (or d.c.c.) on the set of submodules {N ? M: M/N ε K}, then each nil subring of End(M_R ) is nilpotent.     

Verallgemeinerung der Sätze von Johnson und Utumi auf Hom

Ohne Zusammenfassung

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On essential extensions of reduced rings and domains

A ring is said to be a left essential extension of a reduced ring (domain) if it contains a left ideal which is a reduced ring (domain) and intersects nontrivially every nonzero twosided ideal of the ring. We prove that every ring which is a left essential extension of a reduced ring is a subdirect sum of rings which are essential extensions of domains, but the converse implication does not hold. We give some applications of this result and discuss several related questions.Received: 6 January 2003     

Source modules of blocks with normal defect groups

We describe source modules of blocks with normal defect groups over arbitrary ground fields. If a defect Brauer pair of a block is also normalized, we show that there is a graded Morita equivalence between the block and its source algebra. The second author acknowledges the support of a Bolyai Fellowship of the Hungarian Academy of Science. Received: 3 April 2006     

Decisive dimension and other related torsion theoretic dimensions

The paper is concerned with the study of the decisive dimension defined on the category of left modules over a ring R. We compare the decisive dimension with the Gabriel dimension and other dimensions recently introduced. We give module theoretic as well as lattice theoretic characterizations of rings with decisive dimension. As an application we obtain characterizations of some classes of rings.     

Torsion theoretic dimensions and relative Gabriel correspondence

Let τ be an hereditary torsion theory. For a ring with τ-Gabriel dimension, we find necessary and sufficient conditions for the existence of a bijective correspondence between the τ-torsionfree injective modules and the τ-closed prime ideals. As an application, new characterizations of fully bounded noetherian rings are obtained.     

Free submodules of indecomposable modules

Let be an artin algebra which has only finitely many isomorphism classes of non-faithful indecomposable modules. We show that large indecomposable -modules contain large free submodules. Received: 12 April 2003     

Prime rings with semigroup generalized identity

We show that a prime ring satisfies a nontrivial semigroup generalized identity if and only if its central closure is a primitive ring with nonzero socle and the associated skew field is a field.     

The K-Theory of C *-Algebras with Finite Dimensional Irreducible Representations

We study the K-theory of unital C^*-algebras A satisfying the condition that all irreducible representations are finite and of some bounded dimension. We construct computational tools, but show that K-theory is far from being able to distinguish between various interesting examples. For example, when the algebra A is n-homogeneous, i.e., all irreducible representations are exactly of dimension n, then K_*(A) is the topological K-theory of a related compact Hausdorff space, this generalises the classical Gelfand-Naimark theorem, but there are many inequivalent homogeneous algebras with the same related topological space. For general A we give a spectral sequence computing K_*(A) from a sequence of topological K-theories of related spaces. For A generated by two idempotents, this becomes a 6-term long exact sequence.     

Induced liftings, exchange rings and semi-perfect algebras

Several important classes of rings can be characterized in terms of liftings of idempotents with respect to various ideals: classical examples are semi-perfect rings, semi-regular rings and exchange rings. We begin with a study of some extensions of the concept of idempotent lifting and prove the generalizations of some classical lifting theorems. Then we describe the method of induced liftings, which allows us to transfer liftings from a ring to its subrings. Using this method we are able to show that under certain assumptions a subring of an exchange ring is also an exchange ring, and to prove that a finite algebra over a commutative local ring is semi-perfect, provided it can be suitably represented in an exchange ring.     

Strong lifting splits

The concept of an enabling ideal is introduced so that an ideal I is strongly lifting if and only if it is lifting and enabling. These ideals are studied and their properties are described. It is shown that a left duo ring is an exchange ring if and only if every ideal is enabling, that Zhou’s δ-ideal is always enabling, and that the right singular ideal is enabling if and only if it is contained in the Jacobson radical. The notion of a weakly enabling left ideal is defined, and it is shown that a ring is an exchange ring if and only if every left ideal is weakly enabling. Two related conditions, interesting in themselves, are investigated: the first gives a new characterization of δ-small left ideals, and the second characterizes weakly enabling left ideals. As an application (which motivated the paper), let M be an I-semiregular left module where I is an enabling ideal. It is shown that m∈M is I-semiregular if and only if m−q∈IM for some regular element q of M and, as a consequence, that if M is countably generated and IM is δ-small in M, then where for each i.     

Coactions on Hochschild homology of Hopf-Galois extensions and their coinvariants

Let B⊆A be an H-Galois extension, where H is a Hopf algebra over a field K. If M is a Hopf bimodule then , the Hochschild homology of A with coefficients in M, is a right comodule over the coalgebra C_H=H/[H,H]. Given an injective left C_H-comodule V, our aim is to understand the relationship between and . The roots of this problem can be found in Lorenz (1994) [15], where and are shown to be isomorphic for any centrally G-Galois extension. To approach the above mentioned problem, in the case when A is a faithfully flat B-module and H satisfies some technical conditions, we construct a spectral sequence

     

Zweidimensionale Quasialgebren mit Nullteilern

We investigate quasi-algebrasF with zero divisors of dimension 2 over a commutative fieldK which have aK-basis 1,j with an idealKj. Assume thatj belongs to the nucleus ofF. Ifj is an idempotent (and can not be replaced by a nilpotent element) thenF is an algebra, i.e. satisfies both distributative laws. Ifj is nilpotent the possibilities forf depend on the solution of a functional equation first studied by Goab and Schinzel for the field of real numbers. We discuss this functional equation in arbitrary locally compact fields.

     

On Faith's correspondence theorem

A simple proof of an extension of Faith's correspondence theorem for projective modules is given for a Morita context (R, V, W, S) in whichVWV=V andWVW=W.Research supported by N.S.E.R.C. (Canada), Grant No. A 8075.     

Finite-dimensional Leavitt path algebras

We classify the directed graphs E for which the Leavitt path algebra L(E) is finite dimensional. In our main results we provide two distinct classes of connected graphs from which, modulo the one-dimensional ideals, all finite-dimensional Leavitt path algebras arise.