Comorphic rings

A ring R is called left comorphic if, for each a ∈ R, there exists b ∈ R such that Ra = l(b) and r(a) = bR. Examples include (von Neumann) regular rings, and Zpn for a prime p and n ≥ 1. One motivation for studying these rings is that the comorphic rings (left and right) are just the quasi-morphic rings, where R is left quasi-morphic if, for each a ∈ R, there exist b and c in R such that Ra = l(b) and l(a) = Rc. If b = c here the ring is called left morphic. It is shown that R is left comorphic if and only if, for any finitely generated left ideal L ⊆ R, there exists b ∈ R such that L = l(b) and r(L) = bR. Using this, we characterize when a left comorphic ring has various properties, and show that if R is local with nilpotent radical, then R is left comorphic if and only if it is right comorphic. We also show that a semiprime left comorphic ring R is semisimple if either R is left perfect or R has the ACC on {r(x) | x ∈ R}. After a preliminary study of left comorphic rings with the ACC on {l(x) | x ∈ R}, we show that a quasi-Frobenius ring is left comorphic if and only if every right ideal is principal; if and only if every left ideal is a left principal annihilator. We characterize these rings as follows: The following are equivalent for a ring R: R is quasi-Frobenius and left comorphic. R is left comorphic, left perfect and right Kasch. R is left comorphic, right Kasch, with the ACC on {l(x) | x ∈ R}. R is left comorphic, left mininjective, with the ACC on {l(x) | x ∈ R}. Some examples of these rings are given.