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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A commutativity theorem for rings


Author: M. Chacron
Journal: Proc. Amer. Math. Soc. 59 (1976), 211-216
MSC: Primary 16A70
DOI: https://doi.org/10.1090/S0002-9939-1976-0414636-1
MathSciNet review: 0414636
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Abstract: Let $R$ be any associative ring. Suppose that for every pair $({a_1},{a_2}) \in R \times R$ there exists a pair $({p_1},{p_2})$ such that the elements ${a_i} - a_i^2{p_i}({a_i})$ commute, where the ${p_i}$’s are polynomials over the integers with one (central) indeterminate. It is shown here that the nilpotent elements of $R$ form a commutative ideal $N$, and that the factor ring $R/N$ is commutative. This result is obtained by the use of the concept of cohypercenter of a ring $R$, which concept parallels the hypercenter of a ring.


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Keywords: Commutator, polynomials, quasi-regular elements, subgroups preserved re quasi-inner automorphisms
Article copyright: © Copyright 1976 American Mathematical Society