AI Summary & Key Takeaways
AI InsightAmong various application given, the following results are proved: if M is a graded faithful multiplication module, then qg (M) is an idempotent graded multiplication ideal of R such that qg (qg (M)) = qg (M), and every graded representable multiplication R-module is finitely generated.
📌 3 Key Takeaways
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 qg (M) of R is defined. It is proved that the graded ideal qg (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 qg (M) is an idempotent graded multiplication ideal of R such that qg (qg (M)) = qg (M), and every graded representable multiplication R-module is finitely generated.
Özet
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 qg (M) of R is defined. It is proved that the graded ideal qg (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 qg (M) is an idempotent graded multiplication ideal of R such that qg (qg (M)) = qg (M), and every graded representable multiplication R-module is finitely generated.