T-ideals and superAzumaya algebras (Q1284251)

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.