# Definition: 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 $x$.

Algorithms: 1
Axioms: 2
Chapters: 3 4 5 6 7
Definitions: 8 9 10 11
Examples: 12
Explanations: 13
Lemmas: 14
Motivations: 15
Proofs: 16 17 18 19 20 21 22 23 24 25 26 27 28 29
Propositions: 30 31
Solutions: 32 33

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### References

#### Bibliography

1. Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001
2. Fischer, Gerd: "Lehrbuch der Algebra", Springer Spektrum, 2017, 4th Edition