Abstract:
The author constructs an infinite series of finite rings $B$, $B^{(m)}$, $m\geqslant2$, which are not embeddable in rings of matrices over commutative rings, and describes their bases of identities and critical rings of the varieties they generate. He shows that finite rings from the ring varieties $\operatorname{var}B$, $\operatorname{var}B^{(m)}$, $m\geqslant2$, $m=(p-1)t+1$, are either representable by matrices over commutative rings or generate the respective varieties. Under a supplementary restriction on a variety $\mathfrak M$ with exponent $p^k$ it is shown that every finite ring from $\mathfrak M$ is representable by matrices over a commutative ring if and only if $\mathfrak M$ does not contain any of the rings $B$, $B^{(m)}$, $m\geqslant2$.
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