Modules satisfying the prime and maximal radical conditions

Abstract

In this paper, we introduce and study $\mathbb P$-radical and $\mathbb{M}$-radical modules over commutative rings. We say that an $R$-module $M$ is \textit{$\mathbb P$-radical} whenever $M$ satisfies the equality $(\sqrt[p]{{\mathcal{P}}M}:M)=\sqrt{{\mathcal{P}}}$ for every prime ideal ${\mathcal{P}}\supseteq Ann({\mathcal{P}}M)$, where $\sqrt[p]{{\mathcal{P}}M}$ is the intersection of all prime submodules of $M$ containing ${\mathcal{P}}M$. Among other results, we show that the class of $\mathbb P$-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 $\mathbb P$-radical. This also holds for any arbitrary module over an Artinian ring. Furthermore, we call an $R$-module $M$ by $\mathbb{M}$-\textit{radical} if $(\sqrt[p]{{\mathcal{M}}M}:M)={\mathcal{M}}$, for every maximal ideal ${\mathcal{M}}$ containing ${\rm Ann\,}(M)$. We show that the conditions $\mathbb P$-radical and $\mathbb{M}$-radical are equivalent for all $R$-modules if and only if $R$ is a Hilbert ring. Also, two conditions primeful and $\mathbb{M}$-radical are equivalent for all $R$-modules if and only if $\mbox{dim\,}(R)=0$. 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 $\mathbb P$-radical modules in the point of algebraic geometry view.