# Definition:Gradation Compatible with Ring Structure

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## 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** 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 $m$**.