Annihilators in finite rings

Question

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 $ann_r(a)$?

Answer 1

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.