On Commutativity and Centrality in Infinite Rings
Communications in Algebra
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
... T R , C R , and P will denote the center, the set of nilpotent elements, the ideal of torsion elements, the commutator ideal and the prime radical, respectively; and if R has 1, U will be the group of units. For any element or subset Y of R, A Y , A r Y , and A Y will denote the left, right, and 2-sided annihilators of Y ; and Y will be the subring generated by Y . The sets D , D r , and D will be the sets of left, right, and 2-sided zero divisors. As usual, the commutator xy − yx is written as x y ; and for S T ⊆ R, the set S T is defined as s t s ∈ S t ∈ T . The symbols and will have their customary meaning-the set of integers and natural numbers, respectively. An element x of R is called periodic if there exist n m ∈ with n > m such that x n = x m . The ring R is called periodic if each of its elements is periodic. As in Babai and Sós (1985) , a subset X of the semigroup S is called a Sidon subset if for every 4-tuple x y z w of elements of X with x y z w ≥ 3, xy = zw. (In Babai and Sós, 1985 , such a set is called a Sidon subset of the first kind.) We shall need the following result, which generalizes a result in Abdollahi and Mohammadi Hassanabadi (2001). Lemma 2.1. If X is any infinite subset of a cancellative semigroup, then X contains an infinite Sidon subset.