Let $k$ and $k'$ and $n_{1},\ldots,n_{k}$ and $m_{1},\ldots,m_{k'}$ be natural numbers. Let $f_{1}\leq \ldots \leq f_{k}$ and $e_{1} \leq \ldots \leq e_{k'}$ be power primes, such that the following equations hold:

$\prod_{j=1}^{j=k} \prod_{i=0}^{n_{j}-1}(f_{j}^{n_{j}}-f_{j}^{i})=\prod_{j=1}^{j=k'} \prod_{i=0}^{m_{j}-1}(e_{j}^{m_{j}}-e_{j}^{i})$

and

$\prod_{j=1}^{j=k}f_{j}^{n_{j}^{2}}=\prod_{j=1}^{j=k'}e_{j}^{m_{j}^{2}}.$

Is it achievable that $k=k'$ and for each $i=1,\ldots,k$ we have $f_{i}=e_{i}$ and $n_{i}=m_{i}$?

Actually, this question arises from the question that, if two finite semisimple ring $R$ and $R'$ have the same number of members and the same number of unit members, then can we say that these rings are isomorphic together?

If there is some counterexample what additional conditions should be employed to obtain that $R\simeq R'$?