Group rings, matrix rings, and polynomial identities
Author:
Elizabeth Berman
Journal:
Trans. Amer. Math. Soc. 172 (1972), 241248
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
(46 #7302), http://dx.doi.org/10.1090/S00029947197203081873
 [2]
Elizabeth
Berman, Tensor products of polynomial identity
algebras, Trans. Amer. Math. Soc. 156 (1971), 259–271. MR 0274515
(43 #278), http://dx.doi.org/10.1090/S0002994719710274515X
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Nathan
Jacobson, Structure of rings, American Mathematical Society
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(36 #5158)
 [4]
Uri
Leron and Amitai
Vapne, Polynomial identities of related rings, Israel J. Math.
8 (1970), 127–137. MR 0269694
(42 #4589)
 [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
(44 #333)
 [6]
Claudio
Procesi and Lance
Small, Endomorphism rings of modules over
𝑃𝐼algebras, Math. Z. 106 (1968),
178–180. MR 0233846
(38 #2167)
 [1]
 Elizabeth Berman, Matrix rings over polynomial identity rings, Trans. Amer. Math. Soc. 172 (1972), 231239. MR 0308187 (46:7302)
 [2]
 , Tensor products of polynomial identity algebras, Trans. Amer. Math. Soc. 156 (1971), 259271. MR 43 #278. MR 0274515 (43:278)
 [3]
 Nathan Jacobson, Structure of rings, rev. ed., Amer. Math Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R. I., 1964. MR 36 #5158. MR 0222106 (36:5158)
 [4]
 Uri Leron and Amitai Vapne, Polynomial identities of related rings, Israel J. Math. 8 (1970), 127137. MR 42 #4589. MR 0269694 (42:4589)
 [5]
 D. S. Passman, Linear identities in group rings, Pacific J. Math. 36 (1971), 457483. MR 0283100 (44:333)
 [6]
 C. Procesi and L. Small, Endomorphism rings of modules over Plalgebras, Math. Z. 106 (1968), 178180. MR 38 #2167. MR 0233846 (38:2167)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197203081848
PII:
S 00029947(1972)03081848
Keywords:
Group rings,
matrix rings,
polynomial identities,
standard identity,
bounded nil rings
Article copyright:
© Copyright 1972 American Mathematical Society
