Graded module - Knowlpedia

A graded module over a graded ring $R=\bigoplus_{n}R_n$ is an $R$ -module $M$ together with a direct-sum decomposition

M=\bigoplus_{n\in\mathbb Z} M_n

(as an internal direct sum of abelian groups) such that $R_i\cdot M_j\subseteq M_{i+j}$ for all $i,j$ . A homomorphism of graded modules is typically required to preserve degree (or have specified degree shift).

Graded modules are the natural linear objects over a graded ring and encode “homogeneous” algebra in commutative algebra and algebraic geometry.

Examples

Any graded ring $R$ is a graded $R$ -module over itself, with the same decomposition $R=\bigoplus R_n$ .
If $R$ is graded and $I\subseteq R$ is a homogeneous ideal, then $R/I$ is a graded module (and graded ring) with $(R/I)_n = R_n/(I\cap R_n)$ .