graded algebra

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.