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 T:V->W is invertible iff V and W have the same dimension and the column vectors representing the image vectors in W of a basis of V form a nonsingular matrix.

Invertibility can be one-sided. By definition, a map f:X->Y is right-invertible iff it admits a right inverse g:Y->X such that f degreesg=id_Y. This occurs iff f is surjective. Left invertibility is defined in a similar way and occurs iff f is injective.

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

See also

Invertible Element, Inverse, Inverse Function, Invertible Knot

This entry contributed by Margherita Barile

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Rowen, L. H. Ring Theory. New York: Academic Press, pp. 1-2, 1991.

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Barile, Margherita. "Invertible." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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