ON REVERSIBILITY RELATED TO IDEMPOTENTS

Abstract

This article concerns a ring property which preserves the reversibility of elements at nonzero idempotents. A ring R shall be said to be quasi-reversible if 0 6= ab ∈ I(R) for a, b ∈ R implies ba ∈ I(R), where I(R) is the set of all idempotents in R. We investigate the quasireversibility of 2 by 2 full and upper triangular matrix rings over various kinds of reversible rings, concluding that the quasi-reversibility is a proper generalization of the reversibility. It is shown that the quasi-reversibility does not pass to polynomial rings. The structure of Abelian rings is also observed in relation with reversibility and quasi-reversibility. 1. Quasi-reversible rings Throughout every ring is an associative ring with identity unless otherwise stated. Let R be a ring. Use I(R), N∗(R), N(R), and J(R) to denote the set of all idempotents, the upper nilradical (i.e., the sum of all nil ideals), the set of all nilpotent elements, and the Jacobson radical in R, respectively. Note N∗(R) ⊆ N(R). Write I(R)′ = {e ∈ I(R) | e 6= 0}. Z(R) denotes the center of R. The polynomial ring with an indeterminate x over R is denoted by R[x]. Z and Zn denote the ring of integers and the ring of integers modulo n, respectively. Let n ≥ 2. Denote the n by n full (resp., upper triangular) matrix ring over R by Matn(R) (resp., Tn(R)), and Dn(R) = {(aij) ∈ Tn(R) | a11 = · · · = ann}. Use Eij for the matrix with (i, j)-entry 1 and zeros elsewhere, and In denotes the identity matrix in Matn(R). Following Cohn [4], a ring R (possibly without identity) is called reversible if ab = 0 for a, b ∈ R implies ba = 0. Anderson and Camillo [1] used the term ZC2 for the reversibility. A ring (possibly without identity) is usually said to be reduced if it has no nonzero nilpotent elements. Many commutative rings are not reduced (e.g., Znl for n, l ≥ 2), and there exist many noncommutative reduced rings (e.g., direct products of noncommutative domains). It is easily checked that the class of reversible rings contains commutative rings Received August 14, 2018; Revised November 6, 2018; Accepted November 21, 2018. 2010 Mathematics Subject Classification. 16U80, 16S36, 16S50.