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 (p√PM:M)=√P for every prime ideal P⊇Ann(PM), where p√PM 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 (p√MM: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
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
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