Definition 3

Tag 05Y1, permanent. Cite it as [Bourbaki, Tag 05Y1].

Algebra, chapter II, § 11. Graded modules and rings

Let $A$ be a graded ring of type $\Delta$ , $(A_\lambda)$ its graduation and $M$ a left (resp. right) $A$ -module; a graduation $(M_\lambda)$ of type $\lambda$ on the additive group $M$ is compatible with the $A$ -module structure on $M$ if

(2)

A_\lambda M_\mu \subset M_{\lambda + \mu} \quad (\text{resp. } M_\mu A_\lambda \subset M_{\lambda + \mu})

for all $\lambda, \mu$ in $\Delta$ . The module $M$ with this graduation is then called a left (resp. right) graded module of type $\Delta$ over the graded ring $A$ .

When the elements of $\Delta$ are cancellable, it follows from (2) and Proposition 1 that the $M_\lambda$ are $A_0$ -modules.

Clearly if $A$ is a graded ring of type $\Delta$ , the left $A$ -module $A_s$ (resp. the right $A$ -module $A_d$ ) is graded of type $\Delta$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Proposition 1