Varieties of associative rings all whose critical rings are arithmetic (Q1390371)

Let \(\mathcal M\) be a variety of associative rings. The author proves the following main theorem: The following conditions are equivalent: 1) the lattice of ideals of any critical ring from \(\mathcal M\) is distributive; 2) the lattice of ideals of any critical ring from \(\mathcal M\) is a chain; 3) \(\mathcal M\) does not contain the rings \(\left(\begin{smallmatrix} 0 &\mathbb{Z}_p &\mathbb{Z}_p\\ 0 &0 &\mathbb{Z}_p\\ 0 &0 &0\end{smallmatrix}\right)\), \(\langle b_1,\ldots,b_p;\;pb_i=0,\;b_i^2=0,\;b_ib_j=b_jb_i\rangle\), \(\langle m;\;p^2m=0,\;m^3=0\rangle\), \(\langle n;\;p^3n=0,\;n^4=0,\;pn^2=0,\;p^2n=n^3\rangle\) for arbitrary primes \(p\) and the rings \(\langle a,b;\;pa=pb=0,\;ab=-ba,\;a^2=b^2=0\rangle\), \(\langle a,b;\;p^2a=0,\;pb=b^2,\;ab=ba=b,\;a^2=a,\;b^3=0\rangle\) for arbitrary primes \(p>2\).