On rings with inverse adjoint semigroups
type: Item
instance of: scholarly article
title: On rings with inverse adjoint semigroups
zbMATH Open document ID: 0945.16017
published in: Southeast Asian Bulletin of Mathematics
publication date: 2000-10-03T00:00:00Z
review text: Let \(R\) be an associative ring. For all \(x,y\in R\) put \(x\circ y=x+y-xy\). Then \((R,\circ)\) is a semigroup, called the adjoint semigroup of \(R\). The operation \(\circ\) is well known in ring theory, e.g., \(R\) is Jacobson radical if and only if \((R,\circ)\) is a group. The author is interested in the case when \((R,\circ)\) is an inverse semigroup. More exactly, he considers the following problem. Let \(R\) be an associative ring, and let \(S\) be a semigroup; when has the semigroup ring \(RS\) an inverse adjoint semigroup? This problem is solved in two cases: for \(S\) commutative and for \(S\) a band (Theorems 1, 2). In both cases the solution is given in terms of the ideal of \(R\) generated by all idempotents of \(R\). The last result of this paper deals with band-graded rings. Suppose \(B\) is a band, and let \(R\) be a \(B\)-graded ring: \(R=\oplus R_b\) (\(b\in B\)) where every \(R_b\) is a subring of \(R\) with identity \(I_b\), \(R_a R_b\subseteq R_{ab}\), and \(I_aI_b=I_{ab}\) for all \(a,b\in B\) (so-called special \(B\)-graded ring). A ring \(L\) is strongly regular if and only if, for each \(x\in L\), there exists \(y\in L\) such that \(x=x^2y\). Suppose \(R=\oplus R_b\) (\(b\in B\)) is a special \(B\)-graded ring. Theorem 3 states that the following conditions are equivalent: (1) \((R,\circ)\) is inverse, (2) \((R,\circ)\) is a union of groups, (3) \(R\) is strongly regular, (4) all the components \(R_b\) are strongly regular and \(B\) is a semilattice.
Mathematics Subject Classification ID: 16N20
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 20M18
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16E50
zbMATH DE Number: 1406415
zbMATH Keywords: adjoint semigroups
zbMATH Keywords: inverse semigroups
zbMATH Keywords: semigroup rings
zbMATH Keywords: idempotents
zbMATH Keywords: band-graded rings
zbMATH Keywords: strongly regular rings
zbMATH Keywords: semilattices
author: A. V. Kelarev
reviewed by: Josif S. Ponizovskii
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1406415