A complete radical formula and 2-primal modules

TL;DR

This paper introduces a complete radical formula for modules over non-commutative rings, generalizing known results from commutative algebra and exploring properties of 2-primal modules.

Contribution

It defines a complete radical formula for non-commutative modules and proves that 2-primal modules over 2-primal rings behave similarly to modules over commutative rings.

Findings

Modules satisfying the complete radical formula are completely semiprime iff they are subdirect products of completely prime modules.

Confirmed that modules over 2-primal rings are 2-primal, supporting the analogy with commutative rings.

Provides examples and properties of modules satisfying the radical formula, extending classical results.

Abstract

We introduce a complete radical formula for modules over non-commutative rings which is the equivalence of a radical formula in the setting of modules defined over commutative rings. This gives a general frame work through which known results about modules over commutative rings that satisfy the radical formula are retrieved. Examples and properties of modules that satisfy the complete radical formula are given. For instance, it is shown that a module that satisfies the complete radical formula is completely semiprime if and only if it is a subdirect product of completely prime modules. This generalizes a ring theoretical result: a ring is reduced if and only if it is a subdirect product of domains. We settle in affirmative a conjecture by Groenewald and the current author given in \cite{2p} that a module over a 2-primal ring is 2-primal. More instances where 2-primal modules behave like modules over commutative rings are given. This is in tandem with the behaviour of 2-primal rings of exhibiting tendencies of commutative rings. We end with some questions about the role of 2-primal rings in algebraic geometry.