نجمه دهقانی

Let R be a ring with unity. We call two R-modules M and N subisomorphic to each other if there exist R-monomorphisms f: M \rightarrow N and g: N \rightarrow M. Analogue to Schroder- Bernstein Theorem, the question of whether two subisomorphic modules are always isomorphic, has been studied by several authors. In general the answer is negative. On the other hand, an armative answer was shown for the class of (quasi-)injective modules by Bumby and for the class of continuous modules by Muller and Rizvi. It is well known that one cannot weaken this beyond thaking M to be quasi-continuous and N to be continuous. A related analogue question is that of d-subisomorphic modules. We say that R-modules M and N are direct summand subisomorphic (or d-subisomorphic for short) if there exist R-monomorphisms f: M \rightarrow N and g: N \rightarrow M such that Imf and Img are direct summands of N and M respectively. We study the question of when two d-subisomorphic modules are also isomorphic? We proved that if M and N are d-subisomorphic R-modules and one of them is either quasi-continuous or directly nite, then M and N are isomorphic. Further applications and consequences will be discused and examples will be provided.