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 $I^{ \circ} = \operatorname{ Hom}_R(I,E)$). 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.
Categories: math.AC math.RA