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?



A weaker version of strongly graded algebras
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
$$
a\_{i+j} = \sum\_k a\_i^{(k)} \cdot (a\_{-i}^{(k)} \cdot a\_{i+j}) \in A\_i \cdot A\_j.
$$


