Graded submodule

Let $𝑀$ be a graded module over a $𝔄$ -graded ring. A submodule $𝑁 \leq 𝑀$ is said to be graded iff module

𝑁 = ⨁ 𝛼 \in 𝔄 𝑁 𝛼

where $𝑁 𝛼 = 𝑁 \cap 𝑀 𝛼$ for $𝛼 \in 𝔄$ . In this case the inclusion $𝜄 : 𝑁 ↪ 𝑀$ is a homomorphism of graded modules and one may construct the quotient graded module.