Definition:Gradation Compatible with Ring Structure
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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 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 ambiguity.
Also see
Homogeneous Elements
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Elements of $S_m$ are known as homogeneous elements of $R$ of degree $m$.
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