Definition:Gradation Compatible with Ring Structure

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

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

Let $(R_n)_{n\in M}$ be an gradation of type $M$ on the additive group of $R$.

The gradation is compatible with the ring structure
 * $\forall m, n \in M : \forall x \in S_m, y \in S_n: x \circ y \in S_{m \cdot n}$

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.

Also see

 * Definition:Graded Ring

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