REVERSIBLE AND PSEUDO-REVERSIBLE RINGS

This article concerns the structure of idempotents in reversible and pseudo-reversible rings in relation with various sorts of ring extensions. It is known that a ring R is reversible if and only if ab ∈ I(R) for a, b ∈ R implies ab = ba; and a ring R shall be said to be pseudoreversible if 0 6= ab ∈ I(R) for a, b ∈ R implies ab = ba, where I(R) is the set of all idempotents in R. Pseudo-reversible is seated between reversible and quasi-reversible. It is proved that the reversibility, pseudoreversibility, and quasi-reversibility are equivalent in Dorroh extensions and direct products. Dorroh extensions are also used to construct several sorts of rings which are necessary in the process. Throughout every ring is an associative ring with identity unless otherwise stated. Let R be a ring. Use I(R), N∗(R), N∗(R), W (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 lower nilradical (i.e., the intersection of all minimal prime ideals), the Wedderburn radical (i.e., the sum of all nilpotent 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 (power series) ring with an indeterminate x over R is denoted by R[x] (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). Given a set S, |S| means the cardinality of S. We use ∏ to denote the direct product of rings. Following Cohn [2], 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 Received October 30, 2018; Revised January 17, 2019; Accepted February 7, 2019. 2010 Mathematics Subject Classification. 16U80, 16S50, 16S36.