COMMUNICATIONS IN ALGEBRA, 13(3), 599-604 (1985) A Note on the Socle of Graded Modules. C. ~Sstssescu University of Bucharest Rournania F. Van Oystaeyen University of Antwerp, UIA Belgium. 0. Introduction. The main result of this note states that the socle of a graded module over a Z-graded ring is again a graded module. The method of proof i s derived from an idea of G.Bergman which he used in [ Z ] to prove that the Jacobson radical of a Z-graded ring i s a graded ideal. A variation on the same theme has been used by the authors in [ 71 where Jacobson radicals and Brown-McCoy radicals of graded modules have been studied. So we now adapt these methods to the study of the socle of graded modules. Some applications of the main result mentioned are hinted at, however, we do not go to deep into these matters. for compactness sake. For a rather extensive survey of graded 'ring theory we refer to (51 , actually our terminology and notations stem from there. 1. Simple Modules and Finite Extensions. Consider associative rings R and S with unit, such that R C S and S is finitely generated as an R-module by R-centralizing elements. Write : S = Z" Rai, hai = a.X for all h E R, i = I, ..., n. i-I 1.1. Lemma. L e t fl be a simple left S-module. Then fl i s a semisirnple of finite length as a left R-module and all simple R-submodules of N are isomorphic. Proof. S e e Lemma 5 of [ 2 1. Copyright O 1985 by Marcel Dekker, lnc.