Modules with supplements

Let M be an iϋ-module and Ncz M. Any Ha M satisfying = M (i) and H'aH,H' + N = M=>H' =H... ... (ii) will be referred to as a supplement of N in M. In general N need not have a supplement in M. A module M will be said to have property (P x ) if every NaM has a supplement in M. If for every AaM,Nc:M with AΛ-N-M, there exists a supplement H of iV in M satisfying Ha A, we say that M has property (P 2 ).Modules with property (P 2 ) play an important role in our study of dual Goldie dimension.In the present paper we determine the class of rings R with the property that every MeR-moά possesses property (P 2 ) These ture out to be left perfect rings.Also the results obtained here throw more light on the differences between corank and P. Fleury's spanning dimension.