ZARISKI-LIKE SPACES OF CERTAIN MODULES

Abstract

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. The primary-like spectrum $Spec_L(M)$ is the collection of all primary-like submodules $Q$ such that $M/Q$ is a primeful $R$-module. Here, $M$ is defined to be RSP if $rad(Q)$ is a prime submodule for all $Qin Spec_L(M)$. This class contains the family of multiplication modules properly. The purpose of this paper is to introduce and investigate a new Zariski space of an RSP module, called Zariski-like space. In particular, we provide conditions under which the Zariski-like space of a multiplication module has a subtractive basis.