Endlichwertige Vielfachenketten in Ringen mit Associativungleichungen
type: Item
instance of: scholarly article
title: Endlichwertige Vielfachenketten in Ringen mit Associativungleichungen
zbMATH Open document ID: 0301.17002
DOI: 10.1007/BF02410607
author: Reinhold Baer
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1975-01-01T00:00:00Z
review text: The following concept of a generalized ring is introduced: \(R\) is a generalized ring if \(R\) satisfies all the axioms of a ring, but the associative law for multiplication is replaced by the following inequalities: \[ x(yR) x(RY) (xy)R+ (xR)y \tag{A} \] \[ (xR)Y+ (Rx)Y cx(RY) + R(xy) \tag{B} \] \[ (xy)R+ (Yx)R c-x(YR) + Y(xR) \text{ for all }x,y \text{ in }R. \tag{C} \] Ordinary associative rings and Lie-rings are generalized rings. Additional examples are constructed as follows: Let \((G,+)\) be an abelian group, \(\Gamma\) a subgroup of \((\operatorname{End}G,+)\). We take \((R,+) = G\oplus\Gamma\), the direct sum of \(G\) and \(\Gamma\), and define multiplication on \(R\) by \[ (g + \sigma)\cdot (g' + \sigma') =g^{\sigma'} + g'^{\sigma},\text{ with }g, g'\text{ in }G,\ a, a'\text{ in }\Gamma. \] Then \(R\) is a generalized ring if and only if for all \(g\) in \(G\), \(\sigma', \sigma''\) in \(\Gamma\) there exists an \(h\) in \(G\), \(\sigma\) in \(\Gamma\) with \[ g^{\sigma'\sigma''} = h^{\sigma''\sigma'} = g^{\sigma\sigma'}. \] In the following we will say `ring' instead of `generalized ring'. A sequence of elements \(r_1, r_2,\ldots\) in a ring \(R\) is called a sequence of multiples if \(r_{i+1}\) is in \(r_iR\) for all \(i\). Two main problems are investigated: 1) To characterize the elements \(r\) in \(R\) with the property that every sequence of multiples beginning with \(r\) contains a finite number of distinct elements only; 2) To describe the ideals \(J\) of \(R\) which have the property that the number of distinct elements in a sequence of multiples contained in \(J\) is bounded. To formulate some of the results we need the following definitions: Let \(R\) be a ring; \(\mathfrak AR = \{x\in R;\ xR= 0\}\); \(\mathfrak{HA}R = \cap X\), taken over all ideals \(X\) of \(R\) with \(\mathfrak A(R/X) =0\); \(rR^\omega = \) subset \(S\) of \(R\) minimal with the property that \(r\) in \(S\), and \(x\) in \(S\) implies \(xt\) in \(S\) for any element \(t\) i…
Mathematics Subject Classification ID: 17A99
Mathematics Subject Classification ID: 17A30
Mathematics Subject Classification ID: 17D99
zbMATH DE Number: 3470624
reviewed by: H. H. Brungs
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf02410607
OpenAlex ID: W1969682104
description: scientific article; zbMATH DE number 3470624