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.
Konstantinos Kanakoglou
- 7.7k
- 2
- 23
- 64