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C-Commutativity

Published online by Cambridge University Press:  09 April 2009

T. Cheatham  and
E. Enochs
T. Cheatham
Affiliation:
Department of Mathematics, Samford University, Birmingham, Alabama 35209, U.S.A.
E. Enochs
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506, U.S.A.
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Abstract

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An associative ring R with identity is said to be c-commutative for c ∈ R if a, b ∈ R and ab = c implies ba = c. Taft has shown that if R is c-commutative where c is a central nonzero divisor]can be omitted. We show that in R[x] is h(x)-commutative for any h(x) ∈ R [x] then so is R with any finite number of (commuting) indeterminates adjoined. Examples adjoined. Examples are given to show that R [[x]] need not be c-commutative even if R[x] is, Finally, examples are given to answer Taft's question for the special case of a zero-commutative ring.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 2 , December 1980 , pp. 252 - 255
Copyright
Copyright © Australian Mathematical Society 1980

References

Chowdhury, Z. (1971), Zero-commutative rings (Ph.D. dissertation, University of Kentucky).Google Scholar
Hemr, S. (1970), ‘Inherited property for a polynomial ring’, Amer. Math. Monthly 77, 315.Google Scholar