Page "Monomorphism" Paragraph 11

Every morphism in a concrete category whose underlying function is injective is a monomorphism ; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily be a monomorphism in the categorical sense.

In the category of sets the converse also holds, so the monomorphisms are exactly the injective morphisms.

The converse also holds in most naturally occurring categories of algebras because of the existence of a free object on one generator.

In particular, it is true in the categories of all groups, of all rings, and in any abelian category.