Definition:Invertible Element

Definition

Let $\left({S, \circ}\right)$ be an algebraic structure which has an identity $e_S$.

If $x \in S$ has an inverse, then $x$ is said to be invertible for $\circ$.

That is, $x$ is invertible iff:

$\exists y \in S: x \circ y = e_S = y \circ x$

Also see

In the context of a ring $\left({R, +, \circ}\right)$, an element that is invertible in the semigroup $\left({R, \circ}\right)$ is called a unit of $\left({R, +, \circ}\right)$.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): Exercise $4.13$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 4$