Graded ring

Let $(𝔄, +)$ be a monoid. A ring $𝑅$ is said to be $𝔄$ -graded if its additive group $𝑅 +$ is the direct sum of abelian groups $𝑅 𝛼$ indexed by $𝛼 \in 𝔄$ such that $𝑅 𝛼 \cdot 𝑅 𝛽 \subseteq 𝑅 𝛼 + 𝛽$ for any $𝛼, 𝛽 \in 𝔄$ . ring Typically $𝔄 = ℤ$ or $𝑀 = ℕ 0$ , but in principle any monoid can be used.