# Definition:Trivial Gradation

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## Definition

Let $\left({R, +, \circ}\right)$ be a ring.

Let $\left({M, \cdot, e}\right)$ be a monoid.

Define a gradation on $R$ as follows:

- $\displaystyle R_e = R$
- $\displaystyle \forall m \in M \setminus \left\{{e}\right\} : R_m = \mathbf 0$

where $\mathbf 0$ is the zero ring.

This is called the **trivial $M$-grading** on $R$.

## Also known as

A **trivial gradation** is also known as a **trivial grading**.