Über die von einem Ideal $I\subset R$ erzeugten $R$-Moduln II

Abstract

Let $(R,\mathfrak{m})$ be a commutative noetherian local ring and $I$ an ideal of $R$. Let $\mathcal{P}$ be the class of all $I$-generated $R$-modules $M$ (i.e. there is an epimorphism $I^{(\Lambda )}\twoheadrightarrow M$) and let $\mathcal{S}$ be the class of all $I^{\circ }$-cogenerated $R$-modules $N$ (i.e. there is a monomorphism $N\hookrightarrow (I^{\circ }{)}^{\Lambda }$ with $I^{\circ }={\mathrm{Hom}}_R(I,E)$). We give a complete description of all injective and flat modules in $\mathcal{P}$ and $\mathcal{S}$. We show that $(\mathcal{S},\mathcal{P})$ forms a dual pair in the sense of Mehdi--Prest(2015) and that $\mathcal{P}$ is always closed under pure submodules. We determine all ideals $I$ for which $\mathcal{P}$ is closed under submodules, $\mathcal{S}$ is closed under factor modules and $\mathcal{P}$ (resp. $\mathcal{S}$) is closed under group extensions. In the last section, we examine the submodules $\gamma (M)=\sum \{U\subset M|U\in \mathcal{P}\}$ and $\kappa (M)=\bigcap \{V\subset M|M/V\in \mathcal{S}\}$ for all $R$-modules $M$, and we specify their explicit structure in special cases.

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