Let $R$ be a finite commutative ring with identity and $A$ an $R$-algebra with basis $\{1, b\_1,\ldots , b\_k\}$. Suppose that the center of $A$ contains the ring $R$ and there are units $u\_{ij}$ in the ring $R$ such that
$$b\_ib\_j=u\_{ij}b\_jb\_i.$$


(An example: $\mathbb Z\_4[i, j, k]$ ={$a+bi+cj+dk\::\;a, b, c, d\in\mathbb Z\_4$} with the condictions $i^2=j^2=k^2=ijk=-1$. This is the ring of quaternions over $\mathbb Z\_4$, the basis is {$1, i, j, k$}.)


My question is: if $a\in A$ is true that the cardinality of the left annihilator $ann\_l(a)$ equals the cardinality of the right annihilator $ann\_r(a)$?



Annihilators in finite rings 
Let $R$ be a finite field of characteristic $\ne 2$. Let $A$ be the subalgebra of $M\_2(R)$ consisting of upper triangular matrices. It has the basis $1,b\_1,b\_2$ with $b\_2$ the diagonal matrix with entries $-1,1$ and $b\_2$ has zeros on the diagonal and $1$ in the upper right corner. Then $b\_1b\_2=-b\_2$ and $b\_2b\_1=b\_2$ so the condition is satisfied.
But with the element $a$ being the diagonal matrix of entries $1,0$ one gets that the left and right annihilators have different dimension.


