Let  be a function defined on a set  and taking values in a set . Then  is said to be an injection (or injective map, or embedding) if, whenever , it must be the case that . Equivalently,  implies . In other words,  is an injection if it maps distinct objects to distinct objects. An injection is sometimes also called one-to-one.

A linear transformation is injective if the kernel of the function is zero, i.e., a function  is injective iff .

A function which is both an injection and a surjection is said to be a bijection.

In the categories of sets, groups, modules, etc., a monomorphism is the same as an injection, and is used synonymously with "injection" outside of category theory.

REFERENCES:

Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, p. 370, 1989.