On exchange rings

A short, elementary proof is given that right exchange rings are left exchange, with an application to exchange rings with one in the stable range.     

On exchange rings with primitive factor rings artinian

Abstract

In this paper we introduce generalized ideal-stable regular rings. It is shown that if a regular ring R is a generalized I-stable ring, then every square matrix over I is the product of an idempotent matrix and an generalized invertible matrix and admits a diagonal reduction by some generalized invertible matrices.     

Related comparability over exchange rings

It is shown that every exchange ring satisfying related comparability is separative. This yields that related comparability over exchange rings is Morita invariant. Also we investigate pseudosimilarity over exchange rings satisfying related comparability.     

Local rings of minimal length

This paper deals with local rings R possessing an m-canonical ideal ω, R⊆ω. In particular those rings such that the length l_R(ω/R) is as short as possible are studied. The same notion for one-dimensional local Cohen-Macaulay rings was introduced and studied with the name of Almost Gorenstein. Some necessary conditions, that become also sufficient under additional hypotheses, are given and examples are provided also in the non-Noetherian case. The case when the maximal ideal of R is stable is also studied.     

Strong separativity over exchange rings

An exchange ring R is strongly separative provided that for all finitely generated projective right R-modules A and B, A ⊕ A ≅ A ⊕ B ⇒ A ≅ B. We prove that an exchange ring R is strongly separative if and only if for any corner S of R, aS + bS = S implies that there exist u, v ∈ S such that au = bv and Su + Sv = S if and only if for any corner S of R, aS + bS = S implies that there exists a right invertible matrix ∈ M ₂(S). The dual assertions are also proved.     

On N0-quasi-continuous exchange rings

An associative ring R with identity is said to have stable range one if for any a,b? R with aR + bR = R, there exists y ? R such that a + by is left (equivalently, right) invertible. The main results of this note are Theorem 2: A left or right continuous ring R has stable range one if and only if R is directly finite (i.e xy = 1 implies yx = 1 for all x,y ? R), Theorem 6: A left or right N ₀o-quasi-continuous exchange ring has stable range one if and only if it is directly finite, and Theorem 12: left or right N ₀-quasi-continuous strongly π-regular rings have stable range one. Theorem 6 generalizes a well-known result of Goodearl [10], which says that a directly finite, right N _o-continuous von Neumann regular ring is unit-regular     

Local rings of countable Cohen-Macaulay type

We prove (the excellent case of) Schreyer's conjecture that a local ring with countable CM type has at most a one-dimensional singular locus. Furthermore, we prove that the localization of a Cohen-Macaulay local ring of countable CM type is again of countable CM type.

     

Local derivations of full matrix rings

Let \({\mathcal{R}}\) be a unital commutative ring and \({\mathcal{M}}\) be a 2-torsion free central \({\mathcal{R}}\) -bimodule. In this paper, for \({n \geqq 3}\), we show that every local derivation from M _n (\({\mathcal{R}}\)) into M _n (\({\mathcal{M}}\)) is a derivation.     

On the normal ideals of exchange rings

An ideal I of a ring R is called normal if all idempotent elements in I lie in the center of R. We prove that if I is a normal ideal of an exchange ring R then: (1) R and R/I have the same stable range; (2) V(I) is an order-ideal of the monoid C(Specc(R), N), where Specc(R) consists of all prime ideals P such that R/P is local.     

Local derivations of subrings of matrix rings

We show that on finite incidence algebras every local derivation is a derivation.     

Power-substitution,exchange rings and unit II-regularity

In this paper,we investigate power-substitution over exchange rings.We show that an exchange ring R satisfies power-substitution if and only if for any regular x ∈ R, there exists a positive integer n such that xI _n is unit πregular in M _n(R).     

Comparability,separativity, and exchange rings

There are several long-standing open problems which ask whether regular rings, and C^?-algebras of real rank zero, satisfy certain module cancellation properties. Ara, Goodearl, O'Meara and Pardo recently observed that both types of rings are exchange rings, and showed that separative exchange rings have these good cancellation properties, thus answering the questions affirmatively in the separative case. In this article, we prove that, for any positive integer s, exchange rings satisfying s-comparability are separative, thus answering the questions affirmatively in the s-comparable case. We also introduce the weaker, more technical, notion of generalized s-comparability, and show that this condition still implies separativity for exchange rings. On restricting to directly finite regular rings, we recover results of Ara, O'Meara and Tyukavkin.     

Finitely generated projective modules over exchange rings

This paper studies finitely generated projective modules over exchange rings. We prove that cancellation holds inp(R), andK _o (R) is completely determined by the continuous maps from the spectrum ofR toZ ifR is an exchange ring andR/J(R) is a ring with central idempotent elements.     

The exchange property for purely infinite simple rings

It is proven that every purely infinite simple ring is an exchange ring. This result is applied to determine those Leavitt algebras that are exchange rings.

     

Local rings whose modules are almost serial

The present paper is a sequel to our previous work on almost uniserial rings and modules, which appeared in the Journal of Algebra in 2016; it studies rings over which every (left and right) module is almost serial. A module is almost uniserial if any two of its submodules are either comparable in inclusion or isomorphic. And a module is almost serial if it is a direct sum of almost uniserial modules. The results of the paper are inspired by a characterization of Artinian serial rings as rings having all left (or right) modules serial. We prove that if R is a local ring and all left R-modules are almost serial then R is an Artinian ring which is uniserial either on the left or on the right. We also produce a connection between local rings having all left and right modules almost serial, local balanced rings studied by Dlab and Ringel and local Köthe rings. Finally we prove Morita invariance of the almost serial property and list some consequences.     

Local rings whose multiplicative group is nilpotent

We prove that the upper central chain of the multiplicative group of a local ring R coincides with the chain of the multiplicative group of terms of the upper central chain of the associated Lie ring of R. Received: 30 January 2002     

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.