Morphic rings and unit regular rings

A ring RR is called left morphic if R/Ra≅l(a) for every a∈Ra∈R. A left and right morphic ring is called a morphic ring. If Mn(R)Mn(R) is morphic for all n≥1n≥1 then RR is called a strongly morphic ring. A well-known result of Erlich says that a ring RR is unit regular iff it is both (von Neumann) regular and left morphic. A new connection between morphic rings and unit regular rings is proved here: a ring RR is unit regular iff R[x]/(xn)R[x]/(xn) is strongly morphic for all n≥1n≥1 iff R[x]/(x2)R[x]/(x2) 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. This places some known examples in a broader context and answers some existing questions.