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On Commutativity and Centrality in Infinite Rings

2007
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Communications in Algebra
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We show that if R is an infinite ring such that XY ∩ YX = for all infinite subsets X and Y , then R is commutative. We also prove that in an infinite ring R, an element a ∈ R is central if and only if aX ∩ Xa = for all infinite subsets X. 1323 1324 ABDOLLAHI ET AL. Theorem 2. An element a of an infinite ring R is central if and only if aX ∩ Xa = for every infinite subset X of R. PRELIMINARIES We begin by disposing of some notational matters. If R is a ring, the symbols Z, N , T R , C R , and P

doi:10.1080/00927870601142397
fatcat:drrc3ssyavgs5hibp4kklzcoym