Study of Subhomomorphic Property to a Ring

Let M and N be two non-zero right R-modules, M is called subhomomorphic to N in case there exist R?homomorphisms f : M ! N, g : N ! M such that gof is non-zero, and M is called strongly subhomomorphic to N in case there exist homomorphisms f : M ! N, g : N ! M such that both fog and gof are non-zero. After establishing some basic properties of (strongly) subhomomorphic to a ring, it is shown that for a nonsingular ring R the class of injective right R-modules are subhomomorphic to R if and only if R is a semisimple ring.