Generalized Morphic Rings and Their Applications

Let R be a ring. An element a in R is called left morphic (Nicholson and Sánchez Campos, 2004a) if l a R/Ra, where l a denotes the left annihilator of a in R. The ring itself is called a left morphic ring if every element is left morphic. Left morphic rings were first introduced by Nicholson and Sánchez Campos (2004a) and were discussed in great detail there and in Nicholson and Sánchez Campos (2004b, 2005). Our focus is on the case that the condition becomes l a R/Rb for some b ∈ R. We say that the ring R is left generalized morphic if every element satisfies this condition. In Section 2, the definition and some general results are given. Examples of left generalized morphic rings include left morphic rings and left p.p. rings. It is shown that a ring R is left generalized morphic if and only if the exactness of 0 → I → RR → RR of left R-modules implies that I is a principal left ideal. It is also shown that, if R is a left P-injective ring, then R is left generalized morphic if and only if R/aR ∗ is cyclic for every torsionless right R-module of the form R/aR with a ∈ R. Let ( ) be the class of P-projective (P-injective) left R-modules. We prove that is a hereditary cotorsion theory if R is left generalized morphic.