A THEOREM ON ARTINIAN-RINGS

In this short paper, we prove that a commutative Noetherian ring R is Artinian if and only if for any nonzero R-module M there is a nonzero linear map from M into F with F flat and if and only if there is a nonzero linear map from M into P with P projective. An example will show that this is not true for noncommutative rings. The motivation of the present discussion is partially from [1] where flat envelopes of modules were studied. It is also motivated by the study of embeddings of modules into projective modules.