[[Ring theory MOC]]
# Graded ring

Let $(\mathfrak{A},+)$ be a [[monoid]].
A [[ring]] $R$ is said to be $\mathfrak{A}$-**graded** if its additive group $R^+$ is the [[Direct product of groups|direct sum of abelian groups]] $R_{\alpha}$ indexed by $\alpha \in \mathfrak{A}$ 
such that $R_{\alpha} \cdot R_{\beta} \sube R_{\alpha+\beta}$ for any $\alpha,\beta \in \mathfrak{A}$. #m/def/ring
Typically $\mathfrak{A} = \mathbb{Z}$ or $M = \mathbb{N}_{0}$, but in principle any monoid can be used.

## Examples

- [[Tensor algebra]]

## Category of graded rings

See [[Category of graded rings]].

## See also

- [[Graded structure]]

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#state/tidy | #lang/en | #SemBr