Definition:Gradation Compatible with Ring Structure

Definition
Let $\left({R, \oplus, \circ}\right)$ be an ring.

A gradation on $\left({R, \oplus, \circ}\right)$ is a direct sum decomposition into additive groups:
 * $\displaystyle R = \sum_{i=1}^n S_i = S_1 \oplus S_2 \oplus \cdots \oplus S_n$

such that:
 * $\forall x \in S_j, y \in S_k: x \circ y \in S_{j + k}$

and so:
 * $S_j S_k \subseteq S_{j+k}$

Homogeneous Elements
Elements of $S_n$ are known as homogeneous elements of $R$ of degree $n$.