TITLE: $\mathbb{Z}$-graded algebras and tensor products
QUESTION [7 upvotes]: Let $A = \bigoplus_{k \in \mathbb{Z}} A_k$ be a not necessarily commutative $\mathbb{Z}$-graded unital algebra over a field $\mathbb{K}$, and assume that it is strongly graded:
$$
A_kA_l = A_{k+l}.
$$
In general can it happen that the multiplication does not give an isomorphism
$$
A_k \otimes_{A_0} A_l \simeq A_{k+l}?
$$
The map will be surjective since we are assuming the gradind to be strong, but will it be injective?

REPLY [8 votes]: No it cannot happen.
And not only for strongly $\mathbb{Z}$-graded rings; this is always the case for any strongly $G$-graded ring, where $G$ is a group. $A_k \otimes_{A_0} A_l \simeq A_{k+l}$ is an isomorphism of $A_0$-bimodules.
(See: Corollary 3.1.2, p.82, from Methods of Graded Rings).