$HSP\not = SHPS$ for commutative rings with identity

Abstract

Let $I$, $H$, $S$, $P$, $P_s$ be the usual operators on classes of rings: $I$ and $H$ for isomorphic and homomorphic images of rings and $S$, $P$, $P_s$ respectively for subrings, direct, and subdirect products of rings. If $\mathcal K$ is a class of commutative rings with identity (and in general of any kind of algebraic structures), then the class $HSP({\mathcal K})$ is known to be the variety generated by the class $\mathcal K$. Although the class $SHPS({\mathcal K})$ is in general a proper subclass of the class $HSP({\mathcal K})$ for many familiar varieties $HSP({\mathcal K})= SHPS({\mathcal K})$. Our goal is to give an example of a class $\mathcal K$ of commutative rings with identity such that $HSP({\mathcal K})\not = SHPS({\mathcal K})$. As a consequence we will describe the structure of two partially ordered monoids of operators.