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

TL;DR

This paper introduces a new class of modules called -radical modules over commutative rings, explores their properties, and constructs a sheaf theory for their spectra, extending classical ring-theoretic concepts.

Contribution

It defines -radical modules, characterizes their properties over various rings, and develops a sheaf construction for their spectra, linking module theory with algebraic geometry.

Findings

-radical modules include free and finitely generated modules.

All projective modules are -radical over domains and Noetherian rings.

In Artinian rings, all modules are -radical, with characterizations for Noetherian rings.

Abstract

The purpose of this paper and its sequel, is to introduce a new class of modules over a commutative ring $R$, called $\mathbb{P}$-radical modules (modules $M$ satisfying the prime radical condition "$(\sqrt[p]{{\cal{P}}M}:M)={\cal{P}}$" for every prime ideal ${\cal{P}}\supseteq {\rm Ann}(M)$, where $\sqrt[p]{{\cal{P}}M}$ is the intersection of all prime submodules of $M$ containing ${\cal{P}}M$). This class contains the family of primeful modules properly. This yields that over any ring all free modules and all finitely generated modules lie in the class of $\mathbb{P}$-radical modules. Also, we show that if $R$ is a domain (or a Noetherian ring), then all projective modules are $\mathbb{P}$-radical. In particular, if $R$ is an Artinian ring, then all $R$-modules are $\mathbb{P}$-radical and the converse is also true when $R$ is a Noetherian ring. Also an $R$-module $M$ is called $\mathbb{M}$-radical if $(\sqrt[p]{{\cal{M}}M}:M)={\cal{M}}$; for every maximal ideal ${\cal{M}}\supseteq {\rm Ann}(M)$. We show that the two concepts $\mathbb{P}$-radical and $\mathbb{M}$-radical are equivalent for all $R$-modules if and only if $R$ is a Hilbert ring. Semisimple $\mathbb{P}$-radical ($\mathbb{M}$-radical) modules are also characterized. In Part II we shall continue the study of this construction, and as an application, we show that the sheaf theory of spectrum of $\mathbb{P}$-radical modules (with the Zariski topology) resembles to that of rings.