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Generalized n-Like Rings and Commutativity

Published online by Cambridge University Press:  20 November 2018

H. G. Moore
H. G. Moore*
Affiliation:
Brigham Young University, Provo, Utah 84602 U.S.A.
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Abstract

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This note continues the investigation of those rings R with unity which also satisfy the polynomial identity B(x, y) = (xy)n -xyn -xny +xy = 0, for some integer n > l. It is shown that when n is an even integer, or when n = 3, such rings are commutative. It is otherwise possible, as is shown by example, for such rings to fail to be commutative, although they are subdirect sums of local rings satisfying the polynomial identity. Each such ring has nilpotent commutator ideal.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 23 , Issue 4 , 01 December 1980 , pp. 449 - 452
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Herstein, I. N., “A Generalization of a Theorem of Jacobson,” Amer. I. Math. 75 (1953) 105-111.Google Scholar
2. Moore, H. G. and Yaqub, Adil, “On the Commutator Ideal of Certain Rings” Port. Math. 29 (1970) 119-124.Google Scholar
3. Moore, H. G. and Yaqub, Adil “Equational Definability of Addition in Certain Rings” Pac. J. of Math. 74 (1978) 407-417.Google Scholar
4. Yaqub, A., “A Generalization of Certain Rings of A. L. Foster,” Canad. Math. Bull. 6 (1963) 55-60.Google Scholar