Malt’tsev–Neumann products of semi-simple classes of associative rings are studied and some conditions which ensure that such a product is again a semi-simple class are obtained. It is shown that both products, ${𝒮}_1\circ {𝒮}_2$ and ${𝒮}_2\circ {𝒮}_1$ of semi-simple classes ${𝒮}_1$ and ${𝒮}_2$ are semi-simple classes if and only if they are equal.

A class ℱ of universal algebras is called a formation if the following conditions are satisfied: 1) Any homomorphic image of A ∈ ℱ is in ℱ; 2) If α₁, α₂ are congruences on A and $A/{\alpha }_i\in ℱ$, i = 1,2, then A/(α₁∩α₂) ∈ ℱ. We prove that any formation generated by a simple algebra with permutable congruences is minimal, and hence any formation containing a simple algebra, with permutable congruences, contains a minimum subformation. This result gives a partial answer to an open problem of Shemetkov and Skiba...