Admitting an inverse. An object that is invertible is referred to as an invertible element in a monoid or a unit ring, or to a map, which admits an inverse map iff it is bijective. In particular, a linear transformation of finite-dimensional vector spaces is invertible iff and have the same dimension and the column vectors representing the image vectors in of a basis of form a nonsingular matrix.

Invertibility can be one-sided. By definition, a map is right-invertible iff it admits a right inverse such that . This occurs iff is surjective. Left invertibility is defined in a similar way and occurs iff is injective.

The distinction between left and right invertibility makes sense as long as the operation involved is noncommutative (like the composition ), hence it can also be applied more generally to noncommutative monoids and unit rings.