Tensor products of polynomial identity algebras

Author:
Elizabeth Berman

Journal:
Trans. Amer. Math. Soc. **156** (1971), 259-271

MSC:
Primary 16.49

DOI:
https://doi.org/10.1090/S0002-9947-1971-0274515-X

MathSciNet review:
0274515

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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.

- I. N. Herstein,
*Noncommutative rings*, The Carus Mathematical Monographs, No. 15, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968. MR**0227205** - Nathan Jacobson,
*Structure of rings*, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR**0222106** - Claudio Procesi and Lance Small,
*Endomorphism rings of modules over ${\rm PI}$-algebras*, Math. Z.**106**(1968), 178–180. MR**233846**, DOI https://doi.org/10.1007/BF01110128

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Keywords:
Polynomial identity,
tensor product,
matrices

Article copyright:
© Copyright 1971
American Mathematical Society