On the socle of a commutative ring and Zariski topology

Abstract

This paper concerns the coincidence of the socle of a semiprimitive ring (or just a reduced one, in some cases) with the intersection of all essential prime, essential minimal prime, or essential maximal ideals of the ring. More precisely, we prove first that the socle of a reduced ring coincides with the intersection of all essential minimal prime ideals if and only if every minimal prime (or, equivalently, every prime) ideal is either essential or it is a direct summand which is also a maximal ideal. Next, we show that the socle of a semiprimitive ring $R$ is equal to the intersection of all essential maximal ideals of $R$ (i.e., ${Soc}_{\mathsf{max}}\left(R\right)$) if and only if the set of isolated points of $Max\left(R\right)$ with the Zariski topology contains no infinite basic open set. Whenever $R$ is a semiprimitive c.a.c. ring, we prove that for every essential ideal $I$ of $R$ containing ${Soc}_{\mathsf{max}}\left(R\right)$, $I∕{Soc}_{\mathsf{max}}\left(R\right)$ is essential in $R∕{Soc}_{\mathsf{max}}\left(R\right)$ if and only if the set of isolated points of $Max\left(R\right)$ is finite. We apply this result to rings of continuous real-valued functions on a topological space.