Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Tensor products of polynomial identity algebras

Author: Elizabeth Berman
Journal: Trans. Amer. Math. Soc. 156 (1971), 259-271
MSC: Primary 16.49
MathSciNet review: 0274515
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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}, \ldots ,{a_n}$ in A, $ 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}$, 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}$ 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.

References [Enhancements On Off] (What's this?)

  • [1] I. N. Herstein, Noncommutative rings, Carus Math. Monographs, no. 151, Math. Assoc. of America; distributed by Wiley, New York, 1968. MR 37 #2790. MR 0227205 (37:2790)
  • [2] Nathan Jacobson, Structure of rings, 2nd ed., Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R. I., 1964. MR 36 #5158. MR 0222106 (36:5158)
  • [3] Claudio Procesi and Lance Small, Endomorphism rings of modules over PI-algebras, Math. 106 (1968), 178-180. MR 38 #2167. MR 0233846 (38:2167)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16.49

Retrieve articles in all journals with MSC: 16.49

Additional Information

Keywords: Polynomial identity, tensor product, matrices
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society