Semisimple Rings Research Articles

Angus Macintyre, Kenneth McKenna, and Lou van den Dries. Elimination of quantifiers in algebraic structures. Advances in mathematics, vol. 47 (1983), pp. 74–87. - L. P. D. van den Dries. A linearly ordered ring whose theory admits elimination of quantifiers is a real closed field. Proceedings of the American Mathematical Society, vol. 79 (1980), pp. 97–100. - Bruce I. Rose.

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On a commutativity theorem for semi-simple rings

In this note a theorem proved by Abu-Khuzam and Yaqub has been improved as follows: let R be a semi-simple ring such that for all x, y in R there exists a positive integer n = n(x, y) for which either (xy)n + (yx)n or (xy)n – (yx)n is central. Then R is commutative.

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.

Decompositions of graphs of modules over semisimple rings

In this paper we show that an R-module graph over a semisimple ring R can be written as a direct sum of graphic submodules that are uniquely determined up to isomorphism type. Moreover, this decomposition enables us to describe the R-module graph in graphic terms as a disjoint union of connected components, each of which consists of a complete directed graph on its vertices together with a set of loops at each vertex, determined by the loops at 0. We also give a graphic version of Maschke's Theorem.

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”

Some model theory of modules. II. on stability and categoricity of flat modules

This is the second part of a study on model theory of modules begun in [RO]. Throughout, I refer to that paper as “Part I”. I observed there a coincidence between some algebraic and logical points of view in the theory of modules, which led to a convenient representation of p.p. definable sets in flat modules (§1); it becomes especially nice in the case of regular rings (cf. Remark 7 in Part I). Using this in the present paper I obtain a simplified criterion for total transcendence and superstability in the case of flat modules. This enables me to give, besides partial results for arbitrary flat modules (§2), a complete description of the stability classes for modules over regular rings (§3). Particularly, it turns out that a totally transcendental module over a regular ring can be regarded as a module over a semisimple ring (cf. Remark 10 in Part I). This is the crucial observation for the examination of categoricity here: By Morley's theorem, a countable ℵ1-categorical theory is totally transcendental. Consequently, an ℵ1-categorical module over a countable regular ring can be regarded as a module over a semisimple ring. That is why I separately treat categoricity of modules over semisimple rings (§4), even without any assumption on the power of the ring. As a consequence I obtain a complete description of ℵ1-categorical modules over countable regular rings (§5). In the investigation presented here the main tool is the technique of idempotents avoiding the commutativity assumption made in the corresponding results of Garavaglia [GA 1, pp. 86–88], who used maximal ideals for that purpose. At the end of the present paper I show how to derive these latter results in our context.On the way I simplify the known criterion for elementary equivalence for modules over regular rings (§3), which simplifies again in case of semisimple rings (§4). Such a criterion is needed in order to construct Vaughtian pairs (in the categoricity consideration) and it turns out to be useful also for another purpose treated in the third part of this series of papers.

The Surgery Semicharacteristic

Given a degree-one normal map ( / , / ) : (M,vM) -> (X, £), C. T. C. Wall defined the associated surgery obstruction a(f, f) e L^JLn^X). This obstruction vanishes if and only if (/, / ) is normally bordant to a simple homotopy equivalence. Using Wall's approach, one must perform preliminary surgery to make / highly connected before the obstruction can be calculated. It is natural to ask for invariants which can be calculated without preliminary surgery. In the even-dimensional case the multisignature gives such an invariant. The purpose of this paper is to present an odd-dimensional normal bordism invariant defined without preliminary surgery. This invariant is analogous to the semicharacteristic bordism invariant introduced by Lee [13]. A justification for the normal bordism invariant is that it gives strong restrictions on the homology of a manifold with a free action of a finite group. In particular, it gives a good explanation of Lee's results on free actions of finite groups on spheres. Let I be a (2n + l)-dimensional Poincare complex with fundamental group n. Suppose A is a semisimple ring with involution a \-> a, for a e A. Then L2n + 1(A) can be identified as a quotient of a subgroup of ^ 0 (^ ) ( §2). Given a homomorphism (Zn,co) -> (A, —), we can regard A as a local coefficient system (that is, a Znmodule). The surgery semicharacteristic is

On semisimple rings that are centralizer near-rings

On semisimple rings that are centralizer near-rings

The computation of odd dimensional projective surgery groups of finite groups

§0. INTRODUCTION THiS PAPER computes the odd dimensional, projective, surgery groups LPn+I(~) for all finite groups 7r. The computat ion is made as follows. We reduce by Dress induction to the case 1r is 2-hyperelementary and here, we express LPn+l(Tr) as a direct sum of class groups C(X) where X ranges over the irreducible characters of ~r. Several precise numerical calculations are deduced from the general computation. One such calculation is the following: If the Schur index of any irreducible character of a 2-hyperelementary subgroup of ,r is trivial then L1e(Tr) -0; this is the case, for example, if either zr itself or, more generally, every 2-hyperelementary subgroup of is a direct product of abelian and dihedral groups. Another precise calculation is the following: If the 2-hyperelementary subgroups of ~r are abelian then L 3 P ( T T ) (Z[2Z) r°-I where r0 is the number of real, irreducible characters of ~-. The latter calculation and a special case of the former have appeared already in Bak [4]. It is worth mentioning that the even dimensional, projective, surgery groups L~',(~') of finite groups 7r are computed in the following papers; in the case that the 2-hyperelementary subgroups of ~ are abelian, they have been computed already in Bak[4 ,7] and Bak-Schar lau[9] and in the general case, they are computed in Kolster [21] and Kolster [21.1]. We describe now the organization and results of the paper. In §l, we recall briefly the required notation from the Ktheory of quadratic forms; in particular, if A is an involution invariant subring of a semisimple ring B with involution and if A = _+ l then fWo~(A) denotes the Witt group of A-forms (equivalently nonsingular min-quadratic modules) on finitely generated, project ive A-modules which become free of even rank over B. We then cite an exact sequence of Witt groups due to Pardon and Ranicki. In §2, we compute the cokernel of the canonical homorphism

On extensions of nilpotent torsion rings by semisimple rings

A class of rings in which each member is the extension of a nilpotent torsion ring by a semisimple semiartinian ring is presented.

On special radicals, supernilpotent radicals and weakly homomorphically closed classes

AbstractIt is proved that a regular essentially closed and weakly homomorphically closed proper subclass of rings consists of semiprime rings. A regular class M defines a supernilpotent upper radical if and only if M consists of semiprime rings and the essential cover Mk of M is contained in the semisimple class S U M. A regular essentially closed class M containing all semisimple prime rings, defines a special upper radical if and only if M satisfies condition (S): every M-ring is a subdirect sum of prime M-rings. Thus we obtained a characterization of semisimple classes of special radicals; a subclas S of rings is the semisimple class of a special radical if and only if S is regular, subdirectly closed, essentially closed, and satisfies condition (S). The results are valid for alternative rings too.

Partial Coxeter functors and right pure semisimple hereditary rings

An important problem in the representation theory of artinian rings is to obtain a charcterization of rings of finite representation type (see [2, 7, 16, 201). We recall that a ring R is of finite representation type if R is both left and right artinian and there is only a finite number of isomorphism classes of indecomposable finitely generated R-modules. The main tools for studying the representation theory of hereditary artinian rings are partial Coxeter functors and Coxeter functors (see [4, 8, 10, 141). They are successfuly applied in the investigation of hereditary rings of finite representation type [8, 9-12, 141. In the present paper we study partial Coxeter functors for hereditary artinian rings and we use them as a tool for studying right pure semisimple hereditary rings (which are close to rings of finite representation type). We recall that for any ring R the right pure global dimension r.P.gl.dim R and the left pure global dimension l.P.gl.dim R are defined [19-21, 261. There is an interesting problem how to characterize rings R with r.P.gl.dim R = 0 which are exactly those right artinian rings R all of whose right R-modules are algebraically compact [ 19, 20, 261 or, equivalently, all of whose right R-modules are direct sums of modules of finite length. A ring R with r.P.gl.dim R = 0 is called right pure semisimple [25-271. It is well known that any ring of finite representation type is both left and right pure semisimple 12, 20, 23, 251 and the converse implication also holds true [2, 15, 201. However, it is still an open question if a right pure semisimple ring R is of finite representation type or, equivalently, if r.P.gl.dim R = 0 implies l.P.gl.dim R = 0 (see [29]). A positive solution of this problem is given by Auslander [3] for artin algebras. He also discusses the problem for arbitrary left artinian rings (see also 1181). We recall that a 195 0021.8693/81/070195~24$02.00/0

Locally linearly compact semisimple rings.

Locally linearly compact semisimple rings.

Completely semisimple ring semigroups

AbstractLet R be a ring in which the multiplicative semigroup is completely semisimple. If R has the maximum (respectively, minimum) condition on principal multiplicative ideals. then R is semiprime artinian (respectively, a direct sum of dense rings of finite-rank linear transformations of vector spaces over division rings).

A commutativity theorem for division rings

The following theorem is proved: Let D be a division ring such that for all x, y in D there exists a positive integer n = n(x, y) for which (xy)n − (yx)n is in the center of D. Then D is commutative. This theorem also holds for semisimple rings.

On & - semisimple rings. A study of the socle of a ring

(1980). On & - semisimple rings. A study of the socle of a ring. Communications in Algebra: Vol. 8, No. 10, pp. 889-909.

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).

Centralizer near-rings that are endomorphism rings

For a finite ring R with identity and a finite unital R-module V the set C ( R ; V ) = { f : V → V | f ( α v ) = α f ( v ) C(R;V) = \{ f:V \to V|f(\alpha v) = \alpha f(v) for all α ∈ R , v ∈ V } \alpha \in R,v \in V\} is the centralizer near-ring determined by R and V. Those rings R such that C ( R ; V ) C(R;V) is a ring for every R-module V are characterized. Conditions are given under which C ( R ; V ) C(R;V) is a semisimple ring. It is shown that if C ( R ; V ) C(R;V) is a semisimple ring then C ( R ; V ) = End R ( V ) C(R;V) = {\text {End}_R}(V) .

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.

Rings with a finite set of nonnilpotents

Let R be a ring and let N denote the set of nilpotent elements of R. Let n be a nonnegative integer. The ring R is called a θn‐ring if the number of elements in R which are not in N is at most n. The following theorem is proved: If R is a θn‐ring, then R is nil or R is finite. Conversely, if R is a nil ring or a finite ring, then R is a θn‐ring for some n. The proof of this theorem uses the structure theory of rings, beginning with the division ring case, followed by the primitive ring case, and then the semisimple ring case. Finally, the general case is considered.