\begin{definition}[Definition:Gradation Compatible with Ring Structure]
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''' {{iff}}
:$\forall m, n \in M : \forall x \in S_m, y \in S_n: x \circ y \in S_{m \cdot n}$
and so:
:$S_m S_n \subseteq S_{m\cdot n}$
\end{definition}