adj. 1. (of a function, mapping, etc.) associating two sets in such a way that different members of the domain are paired with different members of the codomain, although not all of the latter need be members of the specified range; thus, in the figure below, T is the codomain and T⁺ is the range, so that the mapping from S to T⁺ (as well as that from S to T) is injective. In some usages, an injective map is called one-to-one, but this term may cause confusion, as it is also used for a bijection. For example, the mapping associating oldest sons to fathers is injective even when the codomain is all men; f(x) = x² is not injective on the reals since f(x) = f(-x), but it is injective on the positive real numbers. Compare bijective, surjective. See also monomorphism.
An injective mapping from S to T⁺. 2. (of a left R-module Q) having the property that whenever there is a left R-module A with a submodule B such that there is a homomorphism f of B into Q, then f can be extended to a homomorphism g of A into Q.