A weaker version of strongly graded algebras

Question

Let $A = \oplus_{i \in \mathbb{Z}} A_i$ be a graded algebra. We say that it is strongly graded if $A_i.A_j = A_{i+j}$, for all $i,j \in \mathbb{Z}$. Can there be existing a graded algebra such that $$ A_i.A_{-i} = A_0, ~~ \forall i \in \mathbb{Z}, $$ but which is yet not strongly graded?

Answer 1

If $A_i \cdot A_{-i} = A_0$ then in particular we can write $$ 1 = \sum_k a_i^{(k)} \cdot a_{-i}^{(k)}, $$ where $a_l^{(k)} \in A_l$. Now taking any $a_{i+j} \in A_{i+j}$ and multiplying it by this equality, we obtain $$ a_{i+j} = \sum_k a_i^{(k)} \cdot (a_{-i}^{(k)} \cdot a_{i+j}) \in A_i \cdot A_j. $$