Definition: Associate

Let $(R,\cdot,+)$ be an integral domain with the multiplicative neutral element $1,$ and let $a,b\in R.$

We call $a$ an associate of $b$ (denoted by $a\sim b$) if and only if: $$a\mid b\wedge b\mid a,$$ i.e. $a$ is a divisor of $b$ and vice versa.

Notes

• From this definition, it should be clear that in $R$, any two associates $a,b$ can differ only by a unit.
• For instance, if $R=\mathbb Z,$ any positive integer and its negative counterpart are associates (the only two units of $\mathbb Z$ are $1$ and $-1.$)
• Another example: if $R=\mathbb Q[X]$ is the ring of polynomials with rational coefficients, then two polynomials are associates if they differ only by a rational number $q\neq 0$ as a factor (the units in $\mathbb Q[X]$ are exactly the constant polynomials being the rational numbers unequal $0$).

Definitions: 1
Lemmas: 2
Proofs: 3 4 5
Propositions: 6

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References

Bibliography

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