Graded identities for T-prime algebras over fields of positive characteristic
type: Item
instance of: scholarly article
title: Graded identities for T-prime algebras over fields of positive characteristic
zbMATH Open document ID: 1015.16023
DOI: 10.1007/BF02785422
author: Plamen Koshlukov
author: Sérgio S. de Azevedo
published in: Israel Journal of Mathematics
publication date: 2003-05-07T00:00:00Z
review text: The general structure theory of T-ideals, as developed by Kemer, is based on a careful study of T-semiprime and T-prime T-ideals and heavily depends on the investigation of both ordinary and super polynomial identities of finite dimensional \(\mathbb{Z}_2\)-graded algebras and their Grassmann enveloping algebras. In the case of characteristic 0, the only cases when the polynomial identities of T-prime algebras are completely understood are the Grassmann algebra \(G\), the \(2\times 2\) matrix algebra \(M_2(K)\) with entries from the base field \(K\), the tensor square \(G\otimes G\) of the Grassmann algebra and the algebra \(M_{1,1}(G)\) of all \(2\times 2\) matrices with Grassmann entries which are even on the main diagonal and odd on the other diagonal. It is known that in characteristic 0 the algebras \(G\otimes G\) and \(M_{1,1}(G)\) have the same identities. The study of super (or \(\mathbb{Z}_2\)-graded) identities is both a problem of independent interest and an attempt to understand better the ordinary polynomial identities of PI-algebras. The algebras \(G\), \(M_2(K)\), \(G\otimes G\) and \(M_{1,1}(G)\) have natural \(\mathbb{Z}_2\)-gradings and their \(\mathbb{Z}_2\)-graded identities in characteristic 0 are known. The purpose of the paper under review is to solve similar problems over infinite fields of odd characteristic \(p\). The main results are the following. The graded polynomial identities of \(M_2(K)\) (where the matrix units \(e_{11},e_{22}\) are even and \(e_{12},e_{21}\) are odd) follow from the known identities \(y_1y_2-y_2y_1\) and \(z_1z_2z_3-z_3z_2z_1\), where \(y_i\) are even and \(z_j\) are odd variables. The graded identities of \(M_{1,1}(G)\) (the even and the odd components are at the same places as those of \(M_2(K)\)) are \(y_1y_2-y_2y_1\) and \(z_1z_2z_3+z_3z_2z_1\). In order to obtain the graded identities of \(G\otimes G\) (the even component is \(G_0\otimes G_0\oplus G_1\otimes G_1\) and the odd one is \(G_0\otimes G_1\oplus G_1\…
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16R40
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16W55
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 16R30
zbMATH DE Number: 1819984
zbMATH Keywords: algebras with polynomial identities
zbMATH Keywords: graded polynomial identities of matrices
zbMATH Keywords: Grassmann algebras
zbMATH Keywords: T-prime algebras
MaRDI profile type: Publication
Wikidata QID: Q122171361
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full work available at URL: https://doi.org/10.1007/bf02785422
OpenAlex ID: W2038490070
description: scientific article; zbMATH DE number 1819984