Bulletin of the
Korean Mathematical Society
BKMS

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Bull. Korean Math. Soc. 2021; 58(6): 1401-1407

Published online November 30, 2021 https://doi.org/10.4134/BKMS.b200962

Copyright © The Korean Mathematical Society.

On graded radically principal ideals

Rashid Abu-Dawwas

Yarmouk University

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 $R[X]$.

Keywords: Graded radical ideals, graded principal ideals, graded radically principal ideals, graded radically principal rings

MSC numbers: Primary 13A02, 16W50

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