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?



$\mathbb{Z}$-graded algebras and tensor products
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](https://www.springer.com/gp/book/9783540207467)).


