A commutativity theorem for rings

Chacron, M.

doi:10.1090/S0002-9939-1976-0414636-1

A commutativity theorem for rings
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by M. Chacron

Proc. Amer. Math. Soc. 59 (1976), 211-216

DOI: https://doi.org/10.1090/S0002-9939-1976-0414636-1

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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.

References

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