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

Authors: Caşu Alexei

### Abstract

In this work (which is a continuation of \cite{kas1,kas2,kas3}) the relations between the class $\mathbb C \mathbb O$ of the closure operators of a module category $R$-Mod and the class $\mathbb P \mathbb R$ of preradicals of this category are investigated. The transition from $\mathbb C \mathbb O$ to $\mathbb P \mathbb R$ and backwards is defined by three mappings $\,\Phi : \mathbb C \mathbb O \to \mathbb P \mathbb R\,$ and $\,\Psi_1, \Psi_2 : \mathbb C \mathbb O \to \mathbb P \mathbb R$. The properties of these mappings are studied. Some monotone bijections are obtained between the preradicals of different types (idempotent, radical, hereditary, cohereditary, etc.) of $\mathbb P \mathbb R$ and the closure operators of $\mathbb C \mathbb O$ with special properties (weakly hereditary, idempotent, hereditary, maximal, minimal, cohereditary, etc.).

Institute of Mathematics and Computer Science