Graded multiplication modules and the graded ideal θg(M)

Let G be a group and let R be a G-graded commutative ring. For a graded R-module M, the notion of the associated graded ideal θg(M) of R is defined. It is proved that the graded ideal θg(M) is important in the study of graded multiplication modules. Among various application given, the following results are proved: if M is a graded faithful multiplication module, then θg(M) is an idempotent graded multiplication ideal of R such that θg(θg(M)) = θg(M) , and every graded representable multiplication R-module is finitely generated.