Graded multiplication modules and the graded ideal \theta_g (M)

Abstract

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 \theta_g (M) of R is defined. It is proved that the graded ideal \theta_g (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 \theta_g (M) is an idempotent graded multiplication ideal of R such that \theta_g (\theta_g (M)) = \theta_g (M), and every graded representable multiplication R-module is finitely generated.