If and is an arbitrary element of the ring , then .

The homomorphism 1l of of E into FT maps every element of S to an invertible element of FT.

The same definition holds in any unital ring or algebra where a is any invertible element.

The irreducible fraction for a given element is unique up to multiplication of denominator and numerator by the same invertible element.

An invertible element of mod m is a natural number n < m such that gcd(n, m) = 1.

Now, an element is invertible if there exists such that

A self-invertible element is an element whose inverse is itself.

A group is a monoid all of whose elements are invertible.

A group is a monoid all of whose elements are invertible.

In a monoid the inverse of a central invertible element is a central element.

A group is a monoid every element of which has an inverse.

If and is an arbitrary element of the ring , then .

A field is a nontrivial ring in which every nonzero element is invertible.

An element which possesses a (left/right) inverse is termed (left/right) invertible.

An element is irreducible if it is neither a unit nor the product of two other non-unit elements.