A PRIME ESSENTIAL RING THAT GENERATES A SPECIAL ATOM

Abstract

A special atom (respectively, supernilpotent atom) is a minimal element of the lattice $\mathbb{S}$ of all special radicals (respectively, a minimal element of the lattice $\mathbb{K}$ of all supernilpotent radicals). A semiprime ring $R$ is called prime essential if every nonzero prime ideal of $R$ has a nonzero intersection with each nonzero two-sided ideal of $R$. We construct a prime essential ring $R$ such that the smallest supernilpotent radical containing $R$ is not a supernilpotent atom but where the smallest special radical containing $R$ is a special atom. This answers a question put by Puczylowski and Roszkowska.