Group rings, matrix rings, and polynomial identities

Author:
Elizabeth Berman

Journal:
Trans. Amer. Math. Soc. **172** (1972), 241-248

MSC:
Primary 16A38

MathSciNet review:
0308184

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Abstract: This paper studies the question, if is a ring satisfying a polynomial identity, what polynomial identities are satisfied by group rings and matrix rings over ? Theorem 2.6. *If is an algebra over a field with at least elements, and satisfies , and is a group with an abelian subgroup of index , then the group ring satisfies , where* . Theorem 3.2. *If is a ring satisfying a standard identity, and is a finite group, then satisfies a standard identity*. Theorem 3.4. *If is an algebra over a field, and satisfies a standard identity, then the -by- matrix ring satisfies a standard identity*. Each theorem specifies the degree of the polynomial identity.

**[1]**Elizabeth Berman,*Matrix rings over polynomial identity rings*, Trans. Amer. Math. Soc.**172**(1972), 231–239. MR**0308187**, 10.1090/S0002-9947-1972-0308187-3**[2]**Elizabeth Berman,*Tensor products of polynomial identity algebras*, Trans. Amer. Math. Soc.**156**(1971), 259–271. MR**0274515**, 10.1090/S0002-9947-1971-0274515-X**[3]**Nathan Jacobson,*Structure of rings*, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, R.I., 1964. MR**0222106****[4]**Uri Leron and Amitai Vapne,*Polynomial identities of related rings*, Israel J. Math.**8**(1970), 127–137. MR**0269694****[5]**D. S. Passman,*Linear identities in group rings. I, II*, Pacific J. Math. 36 (1971), 457–483; ibid.**36**(1971), 485–505. MR**0283100****[6]**Claudio Procesi and Lance Small,*Endomorphism rings of modules over 𝑃𝐼-algebras*, Math. Z.**106**(1968), 178–180. MR**0233846**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0308184-8

Keywords:
Group rings,
matrix rings,
polynomial identities,
standard identity,
bounded nil rings

Article copyright:
© Copyright 1972
American Mathematical Society