T-ideals and superAzumaya algebras
type: Item
instance of: scholarly article
title: T-ideals and superAzumaya algebras
zbMATH Open document ID: 0936.16018
author: Louis Rowen
author: Allan Berele
published in: Journal of Algebra
publication date: 2000-01-09T00:00:00Z
review text: Let \(A=A_0\oplus A_1\) be a \(2\)-graded (associative) algebra over a fixed field \(F\), and denote by \(E=E_0\oplus E_1\) the infinite dimensional Grassmann algebra with its natural \(2\)-grading. One considers the algebra \(A^*=A_0\otimes E_0\oplus A_1\otimes E_1\). It is well-known that \(A\) and \(A^{**}\) satisfy the same \(2\)-graded identities, and if \(A\) and \(B\) satisfy the same \(2\)-graded identities then \(A^*\) and \(B^*\) also do. The identities of \(A\) and of \(A^*\) are closely related, see \textit{A. R. Kemer}, Ideals of identities of associative algebras [Transl. Math. Monogr. 87, AMS, Providence RI (1981; Zbl 0732.16001)]. The \(*\) operation plays a very important role in the PI theory. The \(2\)-graded algebra \(A\) satisfies the property (P) if for every (proper) graded ideal \(I\) of \(A\), the graded identities of \(A\) and of \(A/I\) are the same. The paper under review describes the property (P) in terms of the structure of \(A\). Thus one of the main results in the paper is Theorem 13; it states that if \(A\) satisfies (P) then it is a superAzumaya algebra of constant rank. If the field is of characteristic \(\neq 2\) then the converse is shown to be true also (Theorem 18). Applications of these results to relatively free algebras are given as well.
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 17A70
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 16S10
zbMATH DE Number: 1271828
zbMATH Keywords: T-ideals
zbMATH Keywords: Azumaya algebras
zbMATH Keywords: graded identities
zbMATH Keywords: T-prime ideals
zbMATH Keywords: Grassmann algebras
zbMATH Keywords: relatively free algebras
zbMATH Keywords: superAzumaya algebras
reviewed by: Plamen Koshlukov
MaRDI profile type: Publication
full work available at URL: jabr.1998.7633
OpenAlex ID: W2051135788
cites work: Lectures on topics in algebraic \(K\)-theory
cites work: Azumaya-like properties of verbally prime algebras
cites work: Generic verbally prime pi-algebras and their gk-dimensions
cites work: Classification theorems for verbally semiprime algebras
cites work: Supertraces and matrices over Grassmann algebras
cites work: On the codimensions of the verbally prime P.I. algebras
cites work: Q4040687
cites work: VARIETIES AND $ Z_2$-GRADED ALGEBRAS
cites work: Remarks on the prime varieties
cites work: The Polynomial Identities of the Grassman Algebra
cites work: Q5735274
cites work: TRACE IDENTITIES AND CENTRAL POLYNOMIALS IN THE MATRIX SUPERALGEBRAS $ M_{n,k}$
cites work: Tensor products of matrix algebras over the Grassmann algebra
cites work: Q4040049
cites work: The Brauer-Wall Group of a Commutative Ring
cites work: Graded Brauer Groups.
DOI: 10.1006/JABR.1998.7633
Recommended article: Graded identities of several tensor products of the Grassmann algebra
Recommended article: On the graded identities of \(M_{1,1}(E)\)
Recommended article: The varieties generated by \(M_{1,1}(E)\) or \(M_2(K)\)
Recommended article: Graded identities for T-prime algebras over fields of positive characteristic
Recommended article: On \(\mathbb Z_2\)-graded identities of the generalized Grassmann envelope of the upper triangular matrices \(UT_{k,l}(F)\).
Recommended article: Magnum P.I
Recommended article: \(\mathbb{Z}\)-graded polynomial identities of the Grassmann algebra
Recommended article: Q4494207
Recommended article: Representability and Specht problem for \(G\)-graded algebras.
Recommended article: Graded identities for tensor products of matrix (super)algebras over the Grassmann algebra
description: scientific article; zbMATH DE number 1271828