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Definition: Unit

Let $(R,\cdot,+)$ be an integral domain with the multiplicative neutral element $1.$ And element $a\in R$ is called a unit of $R,$ if $$a\mid 1\,$$ i.e. $a$ is a divisor of $1$.

Notes

• Unfolding the definition of a divisor in a ring, this means that there exists \(b\in R\) with \(a\cdot b=1\).
• In other words, units in \(R\) are exactly those of its elements, which have inverse elements with respect to the operation "\(\cdot\)".

Examples

• $1,-1$ are the only units of $\mathbb Z.$
• The units of the polynomial ring $\mathbb Q[X]$ are the (obviously) all non-zero constant polynomials, i.e. the rational numbers $q\in \mathbb Q$ having a multiplicative inverse.

Table of Contents

1. Proposition: Group of Units

Mentioned in:

Definitions: 1 2 3
Proofs: 4 5 6 7 8 9
Propositions: 10 11

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References

Bibliography

1. Koch, H.; Pieper, H.: "Zahlentheorie - Ausgewählte Methoden und Ergebnisse", Studienbücherei, 1976