Let $(X,\ast)$ be an algebraic structure and let $x\in X$. If $e$ denotes the neutral element, an element $y\in X$ with

- $y\ast x=e$ is called
**left-inverse**to $x,$ and - $x\ast y=e$ is called
**right-inverse**to $x.$

If $y$ is both, left-inverse and right-inverse, it is called an **inverse** element of $x$. If such $y$ exists, then we call the element $x$ **invertible**.

- If $”\ast”$ is interpreted as a
**multiplication**operation then we write $x^{-1}$ or $\frac 1x$ instead of $y$ and call $x^{-1}$ the**multiplicative inverse**of $x.$ - If $”\ast”$ is interpreted as an
**addition**operation then we write $-x$ instead of $y$ and call $-x$ the**additive inverse**of $x.$

| | | | | created: 2014-06-08 22:39:00 | modified: 2020-06-27 08:20:22 | by: *bookofproofs* | references: [577], [7896]

[7896] **Fischer, Gerd**: “Lehrbuch der Algebra”, Springer Spektrum, 2017, 4. Auflage

[577] **Knauer Ulrich**: “Diskrete Strukturen – kurz gefasst”, Spektrum Akademischer Verlag, 2001