Modules Satisfying the Prime Radical Condition and a Sheaf Construction for Modules II

TL;DR

This paper extends the theory of $ ext{P}$-radical modules by exploring their spectrum with Zariski topology, analyzing their behavior under localization and sums, and constructing a sheaf structure analogous to algebraic geometry.

Contribution

It develops a sheaf construction for $ ext{P}$-radical modules and shows their spectrum's properties resemble those of rings, advancing module theory and algebraic geometry connections.

Findings

Spectrum of $ ext{P}$-radical modules has a Zariski topology similar to rings.

Behavior of $ ext{P}$-radical modules under localization and direct sums is characterized.

A structure sheaf on $ ext{Spec}(M)$ generalizes the classical sheaf of rings.

Abstract

In this paper we continue our study of modules satisfying the prime radical condition ($\mathbb{P}$-radical modules), that was introduced in Part I (see \cite{BS}). Let $R$ be a commutative ring with identity. The purpose of this paper is to show that the theory of spectrum of $\mathbb{P}$-radical $R$-modules (with the Zariski topology) resembles to that of rings. First, we investigate the behavior of $\mathbb{P}$-radical modules under localization and direct sums. Finally, we describe the construction of a structure sheaf on the prime spectrum Spec$(M)$, which generalizes the classical structure sheaf of the ring $R$ in Algebraic Geometry to the module $M$.