# Radical factorization in finitary ideal systems

## TL;DR

This paper explores radical factorization within finitary ideal systems of cancellative monoids, providing new characterizations and conditions for various classes of monoids and their associated rings.

## Contribution

It introduces novel characterizations for r-almost Dedekind r-SP-monoids and describes conditions for t- and w-SP-monoids, including their impact on monoid rings and *-Nagata rings.

## Key findings

- A monoid is a w-SP-monoid iff the radical of every nontrivial principal ideal is t-invertible.
- Characterization of when monoid rings are w-SP-domains.
- Conditions under which *-Nagata rings are SP-domains for finite type star operations.

## Abstract

In this paper we investigate the concept of radical factorization with respect to finitary ideal systems of cancellative monoids. We present new characterizations for r-almost Dedekind r-SP-monoids and provide specific descriptions of t-almost Dedekind t-SP-monoids and w-SP-monoids. We show that a monoid is a w-SP-monoid if and only if the radical of every nontrivial principal ideal is t-invertible. We characterize when the monoid ring is a w-SP-domain and describe when the *-Nagata ring is an SP-domain for a star operation * of finite type.