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 be any associative ring. Suppose that for every pair there exists a pair such that the elements commute, where the 's are polynomials over the integers with one (central) indeterminate. It is shown here that the nilpotent elements of form a commutative ideal , and that the factor ring is commutative. This result is obtained by the use of the concept of *cohypercenter* of a ring , which concept parallels the hypercenter of a ring.

**[1]**M. Chacron and G. Thierrin,*An algebraic dependence over the quasi-centre*, Ann. Math. Pura Appl. (to appear). MR**0422351 (54:10341)****[2]**M. Chacron,*On a theorem of Herstein*, Canad. J. Math.**21**(1969), 1348-1353. MR**41**#6905. MR**0262295 (41:6905)****[3]**I. N. Herstein,*The structure of a certain class of rings*, Amer. J. Math.**75**(1953), 864-871. MR**15**, 392. MR**0058580 (15:392a)****[4]**-,*On the hypercenter of a ring*, J. Algebra (to appear). MR**0371962 (51:8179)****[5]**-,*A commutativity theorem*, J. Algebra**38**(1976), 112-118. MR**0396687 (53:549)**

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Additional Information

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

Keywords:
Commutator,
polynomials,
quasi-regular elements,
subgroups preserved re quasi-inner automorphisms

Article copyright:
© Copyright 1976
American Mathematical Society