Definition 111.26.3. A *graded $A$-algebra* is a graded $A$-module $B = \bigoplus _{n \geq 0} B_ n$ together with an $A$-bilinear map

that turns $B$ into an $A$-algebra so that $B_ n \cdot B_ m \subset B_{n + m}$. Finally, a *graded module $M$ over a graded $A$-algebra $B$* is given by a graded $A$-module $M$ together with a (compatible) $B$-module structure such that $B_ n \cdot M_ d \subset M_{n + d}$. Now you can define *homomorphisms of graded modules/rings*, *graded submodules*, *graded ideals*, *exact sequences of graded modules*, etc, etc.

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