Definition:Gradation Compatible with Ring Structure

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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 if and only if

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

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

Homogeneous Elements

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