Abstract:A notion of hereditarity of a closure operator with respect to a class of monomorphisms is introduced. Let $C$ be a regular closure operator induced by a subcategory $\Cal A$. It is shown that, if every object of $\Cal A$ is a subobject of an $\Cal A$-object which is injective with respect to a given class of monomorphisms, then the closure operator $C$ is hereditary with respect to that class of monomorphisms.
Keywords: closure operator, hereditary closure operator, injective object, factorization pair
AMS Subject Classification: 18A32, 18G05, 18A20