Semisimple Artinian Ring Research Articles

The moore-penrose inverse of a matrix over a semi-simpie artinian ring

Given an m-by-n matrix over a semi-simple artinian ring with aninvolutory automorphism, necessary and sufficient conditions are given for the matrix tao have a Moore—Penrose inverse. Moreover, expressions for the MP-inverse are obtained. These formulas may be considered as a generalization of the MacDuffee formula for the MP-inverse of a matrix over the complex numbers.

Rings of finite representation type and modules of finite Morley rank

This paper deals with rings whose modules have very pleasant decomposition properties: the right pure semisimple rings. The category of right modules over such a ring is, in many ways, similar to the category of modules over a semisimple artinian ring. Yet it is still an open question whether or not pure semisimplicity actually is a two-sided property (and hence [30] coincides with being of finite representation type). Most of the arguments in this paper are model-theoretic-algebraic in nature: it seems that ideas from model theory, and from the model theory of modules, find natural application in the context considered here. The paper is organized into three sections. The first is introductory and with it I attempt to make the other sections accessible to a reasonably wide readership. In the second section are gathered together various equivalents to right pure semisimplicity. Their equivalence is given a comparatively short, unified, essentially model-theoretic-algebraic proof. The main theorem, in the third section, is that a ring is of finite representation type if and only if all its (right) modules have finite Morley rank. A number of related results are developed. Most of these involve some model theory in their statement, but many are, in essense, algebraic. Since a good deal of information has been compressed into the introductory section, this section should, perhaps, first be read through quickly, and then referred back to as the need arises. This section also is meant to serve as background to a sequel, in which I consider pp formulas and types in terms of the corresponding matrices, and in which the main aim is more purely algebraic. The global conventions are: R denotes a ring with identity; “module”

Rings generated by their regular elements

The units of a ring R are defined by means of a multiplicative property, but in many cases they generate R additively. For example, it is shown in [5, Proposition 6] that if R is a semi-simple Artinian ring then every element of R is a sum of units if and only if the ring S = ℤ/2ℤ⊕ℤ/2ℤ is not a direct summand of R, where ℤ denotes the ring of integers. The theme of this paper is to investigate the corresponding situation concerning regular elements, i. e. elements which are not zero-divisors. We show that if R is a semi-prime right Goldie ring then every element of R is a sum of regular elements if and only if R does not have the ring S defined above as a direct summand (Corollary 2.9). We also characterise those Noetherian rings R such that every element of R is a sum of regular elements (Theorem 2. 6). The characterisation is in terms of the nature of certain prime factor rings of R, and it is again the presence of the ring S, this time in a particular way as a factor ring of R, which prevents R from being generated by its regular elements. If R has no non-zero Artinian one-sided ideals or if 2 is a regular element of R, then every element of R is a sum of regular elements (Corollaries 2. 5 and 2. 7). As an application we show in Section 3 that, for many Noetherian rings R, the set of elements of R which are divisible by every regular element of R is a two-sided ideal of R.

INTEGRALLY CLOSED RINGS

This paper studies integrally closed rings. It is shown that a semiprime integrally closed Goldie ring is the direct product of a semisimple artinian ring and a finite number of integrally closed invariant domains that are classically integrally closed in their (division) rings of fractions. It is shown also that an integrally closed ring has a classical ring of fractions and is classically integrally closed in it. Next, integrally closed noetherian rings are considered. It is shown that an integrally closed noetherian ring all of whose nonzero prime ideals are maximal is either a quasi-Frobenius ring or a hereditary invariant domain. Finally, those noetherian rings all of whose factor rings are invariant are described, and the connection between integrally closed rings and distributive rings is examined. Bibliography: 13 titles.

Polynomes de ore, series formelles tordues et anneaux filters complets herediataires

The Ore extension A [t ; τ, δ] of a semi-simple artinian ring A ,with , is right hereditary, even if the ring monomorphism τ is not surj ective (th . 2.1) . If δ = 0, then R is the associated graded ring of the skew power series ring , so we are led to study fil tered modules over a complete filtered ring A , with decreasing filtrations, which are projective, with respect of a certain family of epimorphisms , in the category of filtered A-modules, which we call f-projective (§§3,4). The skew power series ring over a semisimple artinian ring is right f-hereditary, i.e. every right ideal is f-projective (th. 4.5).

On regular rings and V-rings

In this note, certain generalisations of strongly regular rings are considered in connection with regular rings andV-rings. The result that strongly regular rings are left (and right)V-rings [11] is extended. A condition for prime leftV-rings to be primitive with non-zero socle is given (this is related to a question ofFisher [7, Problem 3]. IfA is an ALD (almost left duo) ring, then (1) a simple leftA-module is injective iff it isp-injective; (2)A is von Neumann regular iff every maximal essential right ideal ofA isf-injective. Characterisations of semi-simple Artinian and simple Artinian rings are given in terms of regular andV-rings.

Semisimple Artinian Rings of Quotients

Semisimple Artinian Rings of Quotients

Semisimple Artinian Rings of Quotients

It is easy to be too busy to pay attention to what anyone else is doing, but not good. All of us should know, and want to know, what has been discovered since our formal education ended, but new words, and relations between them, are growing too fast to keep up. It is possible for a person to learn of the title of a recent work and of the key words used in it and still not have the faintest idea of what the subject is. Progress Reports is to be an almost periodic column intended to increase everyone's mathematical information about what others have been up to. Each column will report one step forward in the mathematics of our time. The purpose is to inform, more than to instruct: what is the name of the subject, what are some of the words it uses, what is a typical question, what is the answer, who found it. The emphasis will be on concrete questions and answers (theorems), and not on general contexts and techniques (theories). References will be kept minimal: usually they will include only one of the earliest papers in which the answer appears and a more recent exposition of the discovery, whenever one is easily available. Everyone is invited to nominate subjects to be reported on and authors to prepare the reports. The ground rules are that the principal theorem should be old enough to have been published in the usual sense of that word (and not just circulated by word of mouth or in preprints); it should be of interest to more than just a few specialists; and it should be new enough to have an effect on the mathematical life of the present and near future. In practice most reports will probably be on progress achieved somewhere between 5 and 15 years ago.

The finite intersection property on annihilator right ideals

In this article the finite intersection property on annihilator right ideals will be shown to be an adequate substitute for more stringent chain conditions on such ideals. One application of the investigation will produce a new characterization of orders in semisimple artinian rings, another will generate new classes of absolutely torsion-free rings.

Rings whose faithful left ideals are cofaithful

A left module M over a ring R is cofaithful in case there is an embedding of R into a finite product of copies of M. Our main result states that a semiprime ring R is left Goldie, that is, has a semisimple Artinian left quotient ring, if and only if R satisfies (i) every faithful left ideal is cofaithful and (ii) every nonzero left ideal contains a nonzero uniform left ideal. The proof is elementary and does not make use of the Goldie and LesieurCroisot theorems. We show that (i) and (ii) are Morita invariant. Moreover, (ii) is invariant under polynomial extensions, and so is (i) for commutative rings. Absolutely torsionfree rings are studied. The ring Q is a left classical quotient ring for the ring R C Q if every regular element (nondivisor of zero) of R is invertible in Q and if every element of Q is of the form b ~ιa where a,b E R and b is regular; in this case we also say that R is a left order in Q. A ring is said to be left Goldie if it has the ascending chain condition on left annihilators and has finite uniform dimension. (A left R -module has finite uniform dimension if it has no infinite direct sum of nonzero submodules, and it is said to be uniform if it is nonzero and any two nonzero submodules have a nontrivial intersection.) A theorem of Goldie [8,9] and Lesieur and Croisot [12] states that a ring is a left order in a semisimple Artinian ring if and only if it is semiprime and left Goldie. It is known that the ascending chain condition on left annihilators is not preserved under an equivalence of categories (Morita invariant); in fact, it does not go up to matrix rings. It is unknown whether being left Goldie is Morita invariant. In section two we give a proof of the theorem stated in the abstract, and in the prime case we give a proof which shows directly that such a ring is an order in a full matrix ring over a division ring. We also weaken the hypothesis of an important theorem on semiprime PI rings. In the third section we use these techniques to study absolutely torsion-free rings. In particular, we show that an absolutely torsionfree ring is Goldie if and only if it has a uniform left ideal, and that the endomorphism ring of a finitely generated projective module over an absolutely torsion-free ring is absolutely torsion-free.

On the existence of generalized inverses

Conditions are given for the existence of a generalized inverse of a ring morphism, and the results are related to the theory of semi-simple Artinian rings. Necessary and sufficient conditions are also given for the existence of a generalized inverse of a morphism on real inner product spaces.

Direct Sums of Torsion-Free Covers

In [4, Theorem 4.1, p. 45], Enochs characterizes the integral domains with the property that the direct product of any family of torsion-free covers is a torsion-free cover. In a setting which includes integral domains as a special case, we consider the corresponding question for direct sums. We use the notion of torsion introduced by Goldie [5]. Among commutative rings, we show that the property “any direct sum of torsion-free covers is a torsion-free cover“ characterizes the semi-simple Artinian rings.

Semiprime modules with maximum conditions

A module R M is semiprime if for each 0 ≠ m ϵ M there exists ƒ ϵ Hom R(M, R) with (mƒ)m ≠ 0. In Section 1 semiprime artinian modules are seen to be isomorphic to finite direct sums of minimal left ideals generated by idempotents. Semiprime noetherian modules have endomorphism rings which are left orders in semisimple artinian rings; and necessary and sufficient conditions for the latter situation to occur are given in Section 3. Prime modules are defined analogously and are treated simultaneously; and the above results are actually considered in the broader milieu of Morita contexts. In Sections 4 and 5 the classical density theorem for rings with faithful minimal left ideals is generalized (with a weakened definition of density) to include semiprime rings possessing faithful finite dimensional left ideals. The method of proof covers the infinite dimensional case as well. As a consequence, the classical density theorem is extended to rings with faithful completely reducible left ideals. In Section 6, the endomorphism ring of a torsionless module over a dense ring of transformations is shown to be a ring of the same type.

Rings of quotients and Morita contexts

The Morita context which has been introduced in [9] was used since to prove Wedderburn theorem on the structure of simple rings [1] and in a disguised form it has been the basic tool in the study of primitive rings with a minimal left ideal (e.g. [6, p. 75]). Other applications, though not stated in an explicit form, can be found in various places. In the present paper we use the Morita context, to obtain various results: Goldie's theorem [2, 3] on the ring of quotients of semi-prime rings, following some ideas of Procesi (e.g. [5, p. 66]) and as a speciliazation one obtains Wedderburn's structure theorems of semi-simple artinian rings. The same methods are very useful in studying the ring of endomorphisms of modules V and in particular we obtain Zelmanowitz result [14] on the structure of Hom( V, V), if V is torsionless and finitely generated, and R is prime or semi-prime. Also, obtained are the facts that ring of endomorphisms of primitive rings are primitive [4, 10] and information on the radicals of ring of endomorphisms.

Rings in which minimal left ideals are projective

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.

On a structure theorem for a semi-simple ring-like object

Every abelian category is an exact category (see, for example, [3], p. 33), however, an exact category is not necessarily an abelian category. The main purpose of this paper is to show that a classical structure theory of semisimple Artinian rings can be generalized in the setting of an exact category. Let E be an exact category and U be an object in E. Let S(U) be the class of subobjects of U. In ([3], p. lo), the intersection and the union of a family (set) in S(U) are defined and it is proven that in an exact category finite union and intersections exist (see [3], Proposition 14.1, p. 16 and Corollary 15.3, p. 19). In other words, if S(U) is a set then S(U) is a lattice. We define a subobject K of an object U in E to be superjIuous [I] in U if, for any subobject A of U with the inclusion v : A -+ U, KU A = U implies that v is an isomorphism. In case the only superfluous subobject in U is zero, we say U is semi-simple. We say an object U in E is a ring-like object provided that (i) U is projective, (ii) [V, I] # $ if 0 # I E S(U), where [V, I] is the set of morphisms from U to 1, (iii) 1r*(1a*1a) =(Ii~I,)~Is for any &ES(U),i= 1,2,3, where Ii * I9 = U~30,V)E[v,ri]X~u,r,~ x&U), /L is the inclusion of Ij to U. Every ring R with 1 is a ring-like object in the category of right R-modules, in which case the product of subobjects is the usual product of right ideals. We shall give, in Section 2, other examples of a ring-like object which are not rings.