Unit-regular Rings Research Articles

K-THEORETICALLY COMPLETE VON NEUMANN REGULAR RINGS

The authors introduce the concept of K0-complete ring and study the properties of K0-complete unit-regular rings.

Rings with the dual of the isomorphism theorem

A ring R satisfies the dual of the isomorphism theorem if R/ Ra≅ l( a) for all elements a of R, where l( a) denotes the left annihilator. We call these rings left morphic. Examples include all unit regular rings and certain left uniserial local rings. We show that every left morphic ring is right principally injective, and use this to characterize the left perfect, right and left morphic rings.

Exchange Rings Satisfying the n-Stable Range Condition, II

This is a continuation of the paper [14]. It is shown that any finite subdirect product of exchange rings satisfying the n-stable range condition is still an exchange ring satisfying the n-stable range condition. Furthermore, we give necessary and sufficient conditions on matrices over an exchange ring R, under which R satisfies the n-stable range condition. This generalizes the corresponding results for unit-regular rings and the stable range one condition.

On Strongly Stable Rings

A ring R is called strongly stable if whenever aR + bR = R, there exists a w ∈ Q(R) such that a + bw ∈ U(R), where Q(R) = {x ∈ R ∣ ∃ e − e 2 ∈ J(R), u ∈ U(R) such that x = eu}. These rings are shown to be a natural generalization of semilocal rings and unit regular rings. We investigate the extensions of strongly stable rings. K 1-groups of such rings are also studied. In this way we recover and extend some results of Menal and Moncasi.

ON PARTIALLY ONE-SIDED UNIT-REGULARITY

ABSTRACT A ring is partially one-sided unit-regular provided that, for each regular , there is a right or left invertible element such that . It is shown that is partially one-sided unit-regular if and only if every regular element in is the product of an idempotent and a right or left invertible element. Morita contexts over partially one-sided unit-regular exchange rings are also studied.

The Nicholson-Varadarajan Theorem on Clean Linear Transformations

An element r in a ring R is clean if r is a sum of a unit and an idempotent. Camillo and Yu showed that unit regular rings are clean and in a very surprising development Nicholson and Varadarajan showed that linear transformations on countable dimension vector spaces over division rings are clean. These rings are very far from being unit regular. Here we note that an idempotent is just a root of g(x)=x^{2}-x . For any g(x) we say R is g(x) -clean if every r in R is a sum of a root of g(x) and a unit. We show that if V is a countable dimensional vector space and over a division ring D and g(x) is any polynomial with coefficients in K={\text Center}D and two distinct roots in K , then {\text End}V_D is g(x) -clean.

UNIT-REGULAR RINGS SATISFYING WEAK COMPARABILITY

We treat unit-regular rings satisfying weak comparability, and show that all matrix rings of factor rings of these rings have the property special (DF). Also a new condition (C) which is related to weak comparability is given, and it is shown that every stably finite regular ring satisfying the condition (C) has the property (DF).

Group actions in a unit-regular ring

Let Rbe a unit-regular ring , let Xbe the set of all nonzero, nonunits of Rand let Gbe the group of all units of R. In this paper, some finiteness properties of Rare investigated by considering group actions of Gon Xas follows:First, in case of half-transitive regualr action if 2 is unit in Ror the number of idempotents in Ris finite, then Ris finite. Secondly, if Gis cyclic and 2 is unit in R, then every orbit under regualr action is a finite set, and so in this case, if Rhas a finite number of idempotents, then Ris finite. Finally, if Fis a field in which 2 is unit and the multiplicative group of all nonzero elenents in Fforms a cyclic group, then Fis finite.

On some properties of simple unit-regular rings

On some properties of simple unit-regular rings

Elements in one-sided unit regular rings

Let R be regular. We show that the following are equivalent:(1) R is a one sided unit regular ring. (2) For every x € R, there exist an idempotente and a right or left invertible u such that x [d] eu or x [d] ue. (3) For every x € R,there exists a right or left invertible u such that xu or ux is an idempotent. Moreover, we give some characterizations of one-sided unit regular rings by group inverses.

On unit-regular rings satisfying s-comparability

On unit-regular rings satisfying s-comparability

Torsion in K0 of unit-regular rings

We construct examples of unit-regular rings R for which K0(R) has torsion, thus answering a longstanding open question in the negative. In fact, arbitrary countable torsion abelian groups are embedded in K0 of simple unit-regular algebras over arbitrary countable fields. In contrast, in all these examples K0(R) is strictly unperforated.

Products of idempotents in regular rings, II

This paper characterizes products of idempotents in (von Neumann) regular rings which are unit-regular or right self-injective. For unit-regular rings, the minimum number of idempotents needed in such a product is determined, thereby generalizing a result of C. S. Ballantine (Products of idempotent matrices, Linear Algebra Appl. 19 (1978), 81–86) in the case of a matrix with entries from a field.

An example of the completion of rank functions over simple unit regular rings

An example of the completion of rank functions over simple unit regular rings

Pseudo-rank functions on skew group rings and on fixed subrings of automorphisms of unit-regular rings

Pseudo-rank functions on skew group rings and on fixed subrings of automorphisms of unit-regular rings

Tensor Products of Dimension Groups and K0 of Unit-Regular Rings

We study direct limits of finite products of matrix algebras (i.e., locally matricial algebras), their ordered Grothendieck groups (K0), and their tensor products. Given a dimension group G, a general problem is to determine whether G arises as K0 of a unit-regular ring or even as K0 of a locally matricial algebra. If G is countable, this is well known to be true. Here we provide positive answers in case (a) the cardinality of G is ℵ1, or (b) G is an arbitrary infinite tensor product of the groups considered in (a), or (c) G is the group of all continuous real-valued functions on an arbitrary compact Hausdorff space. In cases (a) and (b), we show that G in fact appears as K0 of a locally matricial algebra. Result (a) is the basis for an example due to de la Harpe and Skandalis of the failure of a determinantal property in a non-separable AF C*-algebra [18, Section 3].

Unit-regular orthodox semigroups

Unit-regular rings were introduced by Ehrlich [4]. They arose in the search for conditions on a regular ring that are weaker than the ACC, DCC, or finite Goldie dimension, which with von Neumann regularity imply semisimplicity. An account of unit-regular rings, together with a good bibliography, is given by Goodearl [5].

TheN *-metric completion of regular rings

We have two main objectives in this paper. Let R be a unit regular ring (the standard reference is [G1]); R admits a pseudo-metric topology, and the completion, denoted S, is called the N*-completion of R. We study the relations between R and S, complementary to a recent article of Goodearl on N*-complete regular rings. The rings R, S are representable as rings of sections of a sheaf-like object, and we clarify and extend this notion. The collection of pseudo-rank functions of R (terms not defined here, will be found in [G1]), denoted IP(R), is a compact convex set, with extreme boundary denoted either OelP(R) or E. We define N* :R--*[0, 1] via N*(r) =sup{P(r)lPslP(R)}. Then N* induces a pseudo-metric topology on R by means of d(r, s)= N*(r-s). This is a metric precisely when N*(r)=0 implies r = 0 ("R is N*-torsion-free"; in [GH2, p. 208], "R has a Hausdorff family of pseudorank functions"); we shall consider only such rings. Chapter 1 deals with properties of the N*-completion of a general (unit regular) ring R, called S. Our main result here (which takes the bulk of the chapter to establish) is that S possesses no new pseudo-rank functions (i.e., the natural map IP(S)~IP(R) is an affine homeomorphism). (This was claimed in [H1, Proposition 15] but the injectivity part of the proof has a gap.) It follows that S is complete in its intrinsic N*-metric and so the recent results of Goodearl I-G2] apply to S. When the appropriate identifications are made, the sup-norm completion of the image of Ko(R) in AfflP(R) [the affine continuous functions on IP(R)] is just Ko(S ). Some consequences of these results include: (i) If K is a metrizable Choquet simplex, there is an N*-complete regular ring S such that Ko(S ) is order-isomorphic to Aft(K). (ii) If(G, u) is an unperforated interpolation group with order unit (see [EHS]), then the sup-norm closure of its image in its natural representation as a group of affine functions on a simplex, is an interpolation group.

Unit-regular rings and simple self-injective rings

Unit-regular rings and simple self-injective rings

A note on the group structure of unit regular ring elements

1* Introduction* It is well-known that [15, 7] a ring R is strongly regular if and only if every aeR is a group member. In this note we shall use the basic theorem for group members in a ring to show locally that a ring element a e R (with unity) is unit regular exactly when there is a unit ue R and a group G in R such that a e uG. Hence unit regular rings are, as it were locally a rotated version of strongly regular rings. We remind the reader that a ring R is called regular if for every aeR, aeaRa; strongly regular if for every aeR, aeaR, and unit regular if for every aeR, there is a unit u e R such that ana = a [3]. Similar definitions hold locally. A ring with unity is called finite if ah = 1 implies ha — 1. Any solution a~ to axa = a is called an inner or 1-inverse of [1], while any solution a to axa = a and xax = x is called a reflexive or 1-2 inverse of a. For idempotents e and / in R, e ~ / denotes the equivalence in