Bull. Korean Math. Soc. Published online August 6, 2019

Juan Huang, Hai-lan JIN, Yang Lee, and Zhelin Piao
Department of Mathematics, Yanbian University, Yanji 133002, China, Department of Mathematics, Yanbian University, Yanji 133002, China and Institute of Basic Science, Daejin University, Pocheon 11159, Korea

Abstract : 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 2 I(R) for a; b 2 R implies ab = ba; and a ring R shall be said to be pseudo-reversible if
0 6= ab 2 I(R) for a; b 2 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, pseudo-
reversibility, 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.

Keywords : pseudo-reversible ring, reversible ring, Dorroh extension, Abelian ring, quasi-reversible ring, direct product, free algebra, matrix ring, polynomial ring