Prime radicals of graded \(\Omega\)-groups
type: Item
instance of: scholarly article
title: Prime radicals of graded \(\Omega\)-groups
zbMATH Open document ID: 1153.08001
author: I. N. Balaba
author: S. A. Pikhtil'kov
author: A. V. Mikhalev
published in: Journal of Mathematical Sciences (New York)
publication date: 2008-10-28T00:00:00Z
review text: A graded \(\Omega \)-group is an additive group \(A\) with a neutral element \(0\) graded by a group \(G\) with a system \(\Omega \) of \(n\)-ary algebraic operations such that for all \(\omega \in \Omega \) we have \((0,0,\dots,0) \omega =0\). The set \(\Omega \) is nonempty and it contains at least one \(n\)-ary operation such that \(n\geqslant 2\). The group \(A\) is a direct sum of normal subgroups \(A_{g}\), where \(g\in G\), which are called homogeneous components. For all \(a_{1}\in A_{g_{1}}\), \(a_{2}\in A_{g_{2}}\), \dots, \(a_{n}\in A_{g_{n}}\) and any \(n\)-ary operation \(\omega \in \Omega \) we have \(( a_{1},a_{2},\dots,a_{n}) \omega \in A_{g_{1}g_{2}\dots g_{n}}\). If, furthermore, for any finite set \(X\subseteq A\), for all \(n\geqslant 2\), \(a_{1},a_{2},\dots,a_{n}\in X\), \(\omega \in \Omega \), the set of elements \((a_{1},a_{2},\dots,a_{n}) \omega \) is finite, then the graded \( \Omega \)-group \(A\) is said to satisfy the finiteness condition. The elements of the set \(h( A) =\bigcup _{g\in G}A_{g}\) are called homogeneous elements of the \(\Omega \)-group \(A\). A nonempty subset \(I\) of a graded \(\Omega \)-group \(A\) is called a graded ideal if it satisfies the following three conditions: (1) \(I\) is a normal subgroup of the additive group \(A\); (2) for every \(n\)-ary operation \(\omega \in \Omega \) and any elements \(a\in I\) and \(x_{1},x_{2},\dots,x_{n}\in A\) we have \(-( x_{1},x_{2},\dots,x_{n}) \omega +(x_{1},\dots,x_{i-1},a+x_{i},x_{i+1},\dots,x_{n}) \omega \in I\), \( i=1,2,\dots,n\); and (3) if an element \(x\in I\) is the sum \( x=x_{g_{1}}+x_{g_{2}}+\dots+x_{g_{n}}\) of homogeneous elements \(x_{g_{i}}\in A_{g_{i}}\), \(g_{i}\in G\), then \(x_{g_{i}}\in I\) for all \(i=1,2,\dots,n\). In the paper the authors consider only graded ideals. If \(I\subseteq A\) is a graded ideal, then the relation \(a\thicksim b\Leftrightarrow a-b\in I\) is compatible with all operations and the factor set \(A/\!\!\thicksim \) is a graded \(\O…
reviewed by: Halina France-Jackson
Mathematics Subject Classification ID: 08A05
Mathematics Subject Classification ID: 16N80
Mathematics Subject Classification ID: 17A60
zbMATH DE Number: 5358148
zbMATH Keywords: graded omega-group
zbMATH Keywords: finiteness condition
zbMATH Keywords: graded ideal
zbMATH Keywords: strictly Engel element
zbMATH Keywords: prime (semiprime) graded omega-group
zbMATH Keywords: graded prime ideal
zbMATH Keywords: solvable (locally solvable, weakly solvable) graded omega-group
zbMATH Keywords: upper (lower) graded weakly solvable radical of a graded omega-group
MaRDI profile type: Publication
full work available at URL: s10958-008-0053-9
OpenAlex ID: W2060278463
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DOI: 10.1007/S10958-008-0053-9
description: scientific article; zbMATH DE number 5358148