Closure operators in the categories of modules. Part IV (Relations between the operators and preradicals)

In this work (which is a continuation of [1–3]) the relations between the class $\mathbb{CO}$ of the closure operators of a module category $R$-Mod and the class $\mathbb{PR}$ of preradicals of this category are investigated. The transition from $\mathbb{CO}$ to $\mathbb{PR}$ and backwards is defined by three mappings $\Phi\colon \mathbb{CO\to PR}$ and $\Psi_1,\Psi_2\colon\mathbb{CO\to PR}$. The properties of these mappings are studied.
Some monotone bijections are obtained between the preradicals of different types (idempotent, radical, hereditary, cohereditary, etc.) of $\mathbb{PR}$ and the closure operators of $\mathbb{CO}$ with special properties (weakly hereditary, idempotent, hereditary, maximal, minimal, cohereditary, etc.).