Rings with the dual of the isomorphism theorem

A ring R satisfies the dual of the isomorphism theorem if R/Ra l(a) for all elements a of R, where l(a) denotes the left annihilator. We call these rings left morphic. Examples include all unit regular rings and certain left uniserial local rings. We show that every left morphic ring is right principally injective, and use this to characterize the left perfect, right and left morphic rings.