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## Definition: Existence of an Inverse Element

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.

### Notes

• 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]