On splitting cotorsion radicals

Abstract:

For a category of modules, the notion, dual to that of a torsion radical, has been called a cotorsion radical. In this paper, the following two properties are examined for a cotorsion radical $\rho$: (1) If $N$ is a submodule of $M$ and $\rho (M) = M$, then $\rho (N) = N$. (2) The exact sequence $0 \to \rho (M) \to M \to M/\rho (M) \to 0$ splits for each module $M$.