Tensor products of polynomial identity algebras
Elizabeth Berman · Transactions of the American Mathematical Society · 1971
We investigate matrix algebras and tensor products of associative algebras over a commutative ring R with identity, such that the algebra satisfies a polynomial identity with coefficients in R . We call A a P. I. algebra over R if there exists a positive integer n and a polynomial f in n noncommuting variables with coefficients in R , not annihilating A , such that for all a 1 , … , a n {a_1}, \ldots ,{a_n} in A , f ( a 1 , … , a n ) = 0 f({a_1}, \ldots ,{a_n}) = 0 . We call A a P-algebra if f is homogeneous with at least one coefficient of 1. We define the docile identity , a polynomial identity generalizing commutativity, in that if A satisfies a docile identity, then for all n , A n {A_n} , the set of n -by- n matrices over A , satisfies a standard identity. We similarly define the unitary identity , which generalizes anticommutativity. Claudio Procesi and Lance Small recently proved that if A is a P. I. algebra over a field, then for all n , A n {A_n} satisfies some power of a standard identity. We generalize this result to P -algebras over commutative rings with identity. It follows that if A is a P -algebra, A satisfies a power of the docile identity.