### Radical Subcategories

Pacific Journal of Mathematics, Vol. 23, No. 1 (1967), 79-89

Abstract

In a categorical setting we generalize the concept of radical as defined for groups and for rings. We define semi-abelian and co-semi abelian categories. Such categories lack the convenient additive structure of the sets of morphisms between two objects, which may be derived from the duality of the axioms for abelian categories, but, for example, the concept of semi-abelian categories permits one to consider the categories of abelian groups, all groups, commutative rings with identity, all rings, rings with minimum condition, Lie algebras and compact Hausdorff spaces with base points and continuous maps under the same categorical formulation. Generalizations of the classical radical properties are proved; for example, the fact that any object in a semi-abelian category is the extension of a radical object by a semi-simple object and the dual statement.