P.I. degree of tensor products of PI-algebras
Let A be an associative algebra with 1 over a fixed field F of characteristic zero and \(M_ n\) be the algebra of all \(n\times n\) matrices over the field F. By n(A) is denoted the maximal integer n for which every polynomial identity of A also is an identity of \(M_ n.\) In this paper the author investigates the dependence between n(A), n(B) and n(A\(\otimes B)\) and proves that for arbitrary PI-algebras A and B the inequality n(A\(\otimes B)\leq 2n(A)n(B)\) holds and this upper bound for n(A\(\otimes B)\) is sharp, where \(A\otimes B\) is the tensor product of A and B taken over F. The lower bound n(A\(\otimes B)\geq n(A)n(B)\) for n(A\(\otimes B)\) which also is sharp has been established by the author [in Vestn. Mosk. Univ., Ser. I 1980, No.2, 47-50 (1980; Zbl 0429.16014)]. In the paper reviewed she gives a characterization of PI-algebras A with the property that for every PI-algebra B the equality \(n(A)n(B)=n(A\otimes B)\) holds.
scientific article; zbMATH DE number 4101395
scientific article