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Graded identities for algebras with elementary gradings over an infinite field

Part of: Basic linear algebra Rings with polynomial identity Rings and algebras with additional structure

Published online by Cambridge University Press:  10 January 2022

Diogo Diniz Open the ORCID record for Diogo Diniz [Opens in a new window] ,
Claudemir Fidelis Open the ORCID record for Claudemir Fidelis [Opens in a new window]  and
Plamen Koshlukov
Diogo Diniz
Affiliation:
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB58429-970, Brazil (claudemir.fidelis@professor.ufcg.edu.br, diogo@mat.ufcg.edu.br)
Claudemir Fidelis
Affiliation:
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB58429-970, Brazil (claudemir.fidelis@professor.ufcg.edu.br, diogo@mat.ufcg.edu.br) Instituto de Matemática e Estatística da Universidade de São Paulo, São Paulo, SP05508-090, Brazil
Plamen Koshlukov
Affiliation:
Department of Mathematics, UNICAMP, 13083-859Campinas, SP, Brazil (plamen@ime.unicamp.br)
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Abstract

Let $F$ be an infinite field of positive characteristic $p > 2$ and let $G$ be a group. In this paper, we study the graded identities satisfied by an associative algebra equipped with an elementary $G$-grading. Let $E$ be the infinite-dimensional Grassmann algebra. For every $a$, $b\in \mathbb {N}$, we provide a basis for the graded polynomial identities, up to graded monomial identities, for the verbally prime algebras $M_{a,b}(E)$, as well as their tensor products, with their elementary gradings. Moreover, we give an alternative proof of the fact that the tensor product $M_{a,b}(E)\otimes M_{r,s}(E)$ and $M_{ar+bs,as+br}(E)$ are $F$-algebras which are not PI equivalent. Actually, we prove that the $T_{G}$-ideal of the former algebra is contained in the $T$-ideal of the latter. Furthermore, the inclusion is proper. Recall that it is well known that these algebras satisfy the same multilinear identities and hence in characteristic 0 they are PI equivalent.

Keywords

graded identitymatrices over Grassmann algebrasPI equivalenceelementary grading

MSC classification

Primary: 15A75: Exterior algebra, Grassmann algebras 16R10: $T$-ideals, identities, varieties of rings and algebras
Secondary: 16R50: Other kinds of identities (generalized polynomial, rational, involution) 16W50: Graded rings and modules
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 1 , February 2022 , pp. 146 - 166
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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