Loading [MathJax]/jax/output/CommonHTML/jax.js
Open Access
WINTER 2014 Modules satisfying the prime and maximal radical conditions
Mahmood Behboodi, Masoud Sabzevari
J. Commut. Algebra 6(4): 505-523 (WINTER 2014). DOI: 10.1216/JCA-2014-6-4-505

Abstract

In this paper, we introduce and study P-radical and M-radical modules over commutative rings. We say that an R-module M is \textit{P-radical} whenever M satisfies the equality (pPM:M)=P for every prime ideal PAnn(PM), where pPM is the intersection of all prime submodules of M containing PM. Among other results, we show that the class of P-radical modules is wider than the class of primeful modules (introduced by Lu \cite{Lu4}). Also, we prove that any projective module over a Noetherian ring is P-radical. This also holds for any arbitrary module over an Artinian ring. Furthermore, we call an R-module M by M-\textit{radical} if (pMM:M)=M, for every maximal ideal M containing Ann(M). We show that the conditions P-radical and M-radical are equivalent for all R-modules if and only if R is a Hilbert ring. Also, two conditions primeful and M-radical are equivalent for all R-modules if and only if dim\,(R)=0. Finally, we remark that the results of this paper will be applied in a subsequent work of the authors to construct a structure sheaf on the spectrum of P-radical modules in the point of algebraic geometry view.

Citation

Download Citation

Mahmood Behboodi. Masoud Sabzevari. "Modules satisfying the prime and maximal radical conditions." J. Commut. Algebra 6 (4) 505 - 523, WINTER 2014. https://doi.org/10.1216/JCA-2014-6-4-505

Information

Published: WINTER 2014
First available in Project Euclid: 5 January 2015

zbMATH: 1309.13012
MathSciNet: MR3294860
Digital Object Identifier: 10.1216/JCA-2014-6-4-505

Subjects:
Primary: 13A99 , 13C13 , 13C99 , 14A25

Keywords: P-radical module , prime spectrum , Prime submodule , sheaf of modules , sheaf of rings , Zariski topology

Rights: Copyright © 2014 Rocky Mountain Mathematics Consortium

Vol.6 • No. 4 • WINTER 2014
Back to Top