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.
