A mathematical object is invertible if it admits an inverse. In a monoid or unit ring, such an object is called an invertible
element. A map is invertible iff
it admits an inverse map, equivalently 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 an invertible 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.
