Regular elements determined by generalized inverses

Communications in Algebra, 2004

All rings R considered are commutative and have an identity element. Contessa called R a VNL-ring if a or 1 À a has a Von Neumann inverse whenever a 2 R. Sample results: Every prime ideal of a VNL-ring is contained in a unique maximal ideal. Local and Von Neumann regular rings are VNL and if the product of two rings is VNL, then both are Von Neumann regular, or one is Von Neumann regular and the other is VNL. The ring Z n of integers mod n is VNL iff ðpqÞ 2 j n whenever p and q are distinct primes. The ring R½½x of formal power series over R is VNL iff R is local. The ring CðXÞ of all continuous real-valued functions on a Tychonoff space X is VNL if and only if at most one point of X fails to be a P-point. All known VNL-rings satisfy SVNL, namely whenever the ideal generated by a (finite) subset of R is all of R; one of its members has a Von Neumann inverse. We show that a ring R is SVNL if and only if all maximal ideals of R are pure except maybe one. We show that Q a2I RðaÞ is an SVNL if and only if there exists a 0 2 I; such that Rða 0 Þ is an SVNL and for all a 2 I À fa 0 g, RðaÞ is a Von Neumann regular ring. Whether every VNL-ring is an SVNL is an open question. # Communicated by W. Martindale.

Journal of Algebra

We study the first-order theory of Bezout difference rings. In particular we show that rings of sequences very rarely have decidable theories as difference rings, or even decidable model completions.

International Journal of Algebra

We study the structure of certain von Neumann regular rings and π-regular rings with certain constraints such as having a prime and other constraints. For example, we prove that a π-regular ring with prime center is strongly π-regular, and other related results are also proved. An example is also given to illustrate our result.

Using the recent notion of inverse along an element in a semigroup, and the natural partial order on idempotents, we study bicommuting generalized inverses and define a new inverse called natural inverse, that generalizes the Drazin inverse in a semigroup, but also the Koliha-Drazin inverse in a ring. In this setting we get a core decomposition similar to the nilpotent, Kato or Mbekhta decompositions. In Banach and Operator algebras, we show that the study of the spectrum is not sufficient, and use ideas from local spectral theory to study this new inverse.

2000

The aim of this paper is to prove that, if R is a commutative regular ring in which 2 is a unit, then the reduced theory of quadratic forms with invertible coe cients in R, modulo a proper preorder T, satisfies Marshall's signature conjecture and Milnor's Witt ring conjecture (for precise statements, see Section 1 below). For that purpose we

Pacific Journal of Mathematics, 1977

Several new properties are derived for von Neumann finite rings. A comparison is made of the properties of von Neumann finite regular rings and unit regular rings, and necessary and sufficient conditions are given for a matrix ring over a regular ring to be respectively von Neumann finite or unit regular. The converse of a theorem of Henriksen is proven, namely that if R n x n , the n x n matrix ring over ring R, is unit regular, then so is the ring R. It is shown that if R 2 2 is finite regular then a e R is unit regular if and only if there is x e R such that R -aRΛ-x(a°), where a 0 denotes the right annihilator of a in R.

Communications in Algebra, 2001

A number of main properties of the commuting regular rings and commuting regular semigroups have been studied in this paper. Some significant results of which will be used for the commutative rings and a necessary and sufficient condition is given for a semigroup to be commuting regular.

Ring Theory 2007, Proceedings, 2009

We survey recent progress on the realization problem for von Neumann regular rings, which asks whether every countable conical refinement monoid can be realized as the monoid of isoclasses of finitely generated projective right R-modules over a von Neumann regular ring R.

In this talk several properties of the inverse along an element in the context of unitary rings will be presented. Among others results, the set of all invertible elements along a fixed element will be fully described. In addition, commuting inverse along a fixed element will be characterized. Furthermore, the special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) will be also considered.