Definition:Gradation Compatible with Ring Structure

Definition
Let $\left({M, \cdot, e}\right)$ be a monoid.

Let $\left({R, +, \circ}\right)$ be a ring.

An $M$-gradation on $\left({R, +, \circ}\right)$ is a direct sum decomposition into additive groups:


 * $\displaystyle R = \bigoplus_{m \mathop \in M} S_m$

such that:


 * $\forall x \in S_m, y \in S_n: x \circ y \in S_{m \cdot n}$

and so:


 * $S_m S_n \subseteq S_{m\cdot n}$

Also known as
An $M$-gradation can also be seen referred to as an $M$-grading.

The terms gradation or grading can also be found when there is no chance of aumbiguity.

Homogeneous Elements
Elements of $S_m$ are known as homogeneous elements of $R$ of degree $m$.