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 ρ \rho : (1) If N N is a submodule of M M and ρ ( M ) = M \rho (M) = M , then ρ ( N ) = N \rho (N) = N . (2) The exact sequence 0 → ρ ( M ) → M → M / ρ ( M ) → 0 0 \to \rho (M) \to M \to M/\rho (M) \to 0 splits for each module M M .