Closure operators in the categories of modules. Part I (Weakly hereditary and idempotent operators)

In this work the closure operators of a category of modules $R$-Mod are studied. Every closure operator $C$ of $R$-Mod defines two functions $\mathcal{F}_1^{C}$ and $\mathcal{F}_2^{C}$, which in every module $M$ distinguish the set of $C$-dense submodules $\mathcal{F}_1^{C}(M)$ and the set of $C$-closed submodules $\mathcal{F}_2^{C}(M)$. By means of these functions three types of closure operators are described: 1) weakly hereditary; 2) idempotent; 3) weakly hereditary and idempotent.