Graded module

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

(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 $(R/I)_n = R_n/(I\cap R_n)$.