On rings whose finitely generated left ideals are left annihilators of an element

Abstract

An associative ring $R$ with identity is left pseudo-morphic if for every $a$$\in$$R$, there exists $b$$\in$$R$ such that $Ra=l_R(b)$. If, in addition, $l_R(a)=Rb$, then $R$ is called left morphic. $R$ is morphic if it is both left and right morphic. We characterize left pseudo-morphic rings; identify the cases a (left) pseudo morphic ring is (left) quasi-morphic, morphic, Quasi-Frobenius, von Neumann regular, etc.; correct two results in a book and a paper; and completely determine when the trivial extension of a commutative domain is morphic which positively answered a question in a paper.

20 pages, 4 figures. Comments are very welcome(email:[email protected]). This is a revised version of Feb 2010 arxiv:1002.3193v1 which only contains section 4 of the present version.

References in corpus (2)

• Morphic and principal-ideal group rings
• A Characterization of Certain Morphic Trivial Extensions