Definition:Gradation Compatible with Ring Structure

Definition
Let $\struct {M, \cdot}$ be a semigroup.

Let $\struct {R, +, \circ}$ be a ring.

Let $\sequence {R_n}_{n \mathop \in M}$ be a 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 ambiguity.

Also see

 * Definition:Graded Ring

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