Matrix rings over polynomial identity rings

Author:
Elizabeth Berman

Journal:
Trans. Amer. Math. Soc. **172** (1972), 231-239

MSC:
Primary 16A42

DOI:
https://doi.org/10.1090/S0002-9947-1972-0308187-3

MathSciNet review:
0308187

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Abstract: We prove that if is an algebra over a field with at least elements, and satisfies , then , the ring of -by- matrices over , satisfies , where . Theorem 1.3 generalizes this result to rings: If is a ring satisfying , then for all , there exists such that satisfies .

Definitions. A *checkered permutation* of the first positive integers is a permutation of them sending even integers into even integers. The *docile polynomial of degree* is

*docile product polynomial of degree*is

*Any polynomial identity algebra over a field of characteristic*0

*satisfies a docile product polynomial identity*. Theorem 2.2.

*If is a ring satisfying the docile product polynomial identity of degree , and is a positive integer, and ; then satisfies a product of standard identities, each of degree*.

**[1]**Elizabeth Berman,*Tensor products of polynomial identity algebras*, Trans. Amer. Math. Soc.**156**(1971), 259–271. MR**0274515**, https://doi.org/10.1090/S0002-9947-1971-0274515-X**[2]**Nathan Jacobson,*Structure of rings*, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, R.I., 1964. MR**0222106****[3]**Neal H. McCoy,*Rings and ideals*, Carus Monograph Series, no. 8, The Open Court Publishing Company, LaSalle, Ill., 1948. MR**0026038****[4]**Claudio Procesi and Lance Small,*Endomorphism rings of modules over 𝑃𝐼-algebras*, Math. Z.**106**(1968), 178–180. MR**0233846**, https://doi.org/10.1007/BF01110128

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DOI:
https://doi.org/10.1090/S0002-9947-1972-0308187-3

Keywords:
Polynomial identity rings,
nil rings,
matrix rings

Article copyright:
© Copyright 1972
American Mathematical Society