Let $R$ be an artinian commutative ring and $A$ an algebra over $R$ with basis $\{a_1, a_2, \ldots , a_n\}$ where each $a_i$ is a unit and

$a_ia_j=u_{ij}a_ja_i,$

where each $u_{ij}\in R$ and is a unit. is true that for every element $x\in A$ the left annihilator of $x$ denoted by $ann_l\{x\}$ equals the right annhilator $ann_r\{x\}$?

I think that if exist an element $z$ with $ann_l{z}\neq ann_r{z}$, this lack of symmetry must necessarily be reflected in the ring $R$ or in the elements of the base. Since, for the ring $R$ and for the elements of the base there is symmetry, this symmetry extends to all elements of $A$. This is my perception, but how to prove?