An algebra $A$ over a graded ring $B$ is graded if it is itself a graded ring and a graded module over $B$ such that $$A^p \cdot A^q \subseteq A^{p+q}$$ where $A^i$ , $i \in \mathbb{N}$ , are submodules of $A$ . More generally, one can replace $\mathbb{N}$ by a monoid or semigroup $G$ . In which case, $A$ is called a $G$ -graded algebra. A graded algebra then is the same thing as an $\mathbb{N}$ -graded algebra.

Examples of graded algebras include the polynomial ring $k[X]$ being an $\mathbb{N}$ -graded $k$ -algebra, and the exterior algebra.