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 $1$.

- 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 "\(\cdot\)".

- $1,-1$ 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.

Definitions: 1 2 3

Proofs: 4 5 6 7 8 9

Propositions: 10 11

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