On Graded Radically Principal Ideals

TL;DR

This paper introduces and studies the concept of graded radically principal ideals in G-graded rings, establishing characterizations and properties, including an analogue of Cohen's theorem and analysis for polynomial extensions.

Contribution

It defines graded radically principal ideals and rings, proves a graded Cohen theorem, and explores properties of polynomial rings over such rings.

Findings

A graded Cohen-like theorem for graded radically principal rings.

Characterization of graded radically principal rings via prime ideals.

Analysis of the graded radically principal property in polynomial rings.

Abstract

Let $R$ be a commutative $G$-graded ring with a nonzero unity. In this article, we introduce the concept of graded radically principal ideals. A graded ideal $I$ of $R$ is said to be graded radically principal if $Grad(I)=Grad(\langle c\rangle)$ for some homogeneous $c\in R$, where $Grad(I)$ is the graded radical of $I$. The graded ring $R$ is said to be graded radically principal if every graded ideal of $R$ is graded radically principal. We study graded radically principal rings. We prove an analogue of the Cohen theorem, in the graded case, precisely, a graded ring is graded radically principal if and only if every graded prime ideal is graded radically principal. Finally we study the graded radically principal property for the polynomial ring $R[X]$.