Invertible

An element $x$ from a unital magma $(M,*)$, with identity $e$, is said to be invertible if there exist $y,z\in M$ such that $x*y=e$ and $z*x=e$.

Remarks

• If the operation $*$ is associative, hence if $(M,*)$ is a monoid, then one can set $y=z$ and say that $y$ is the inverse of $x$, sometimes denoted as $x^{*-1}$. It is necessarily unique in this case.

• In the case of a ring $(R,+,\times)$, then $(R,\times)$ is a monoid hence each invertible element has to be unique. We denote by $x^{-1}$ the inverse of $x$ if it exists.