Morphic rings and unit regular rings. (Q886234)

A ring \(R\) is called left morphic if \(R/Ra\simeq\mathbf l(a)\) for every \(a\in R\). A left and right morphic ring is called a morphic ring. If \(\mathbb{M}_n(R)\) is morphic for all \(n\geq 1\) then \(R\) is called a strongly morphic ring. A well-known result of Erlich says that a ring \(R\) is unit regular iff it is both (von Neumann) regular and left morphic. A new connection between morphic rings and unit regular rings is obtained in this paper: a ring \(R\) is unit regular iff \(R[x]/(x^n)\) is strongly morphic for all \(n\geq 1\) iff \(R[x]/(x^2)\) is morphic. Various new families of left morphic or strongly morphic rings are constructed as extensions of unit regular rings and of principal ideal domains in this paper. This places some known examples in a broader context and answers some existing questions.