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Definition: Generalization of Divisor and Multiple

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

We call $a$ a divisor of $b$ (denoted by $a\mid b$), if and only if there exists an element $c\in R$ such that $ac=b.$ If $a\mid b,$ then we call $b$ a multiple of $a.$

Notes

• This definition corresponds to the special case of a divisor, if $R=\mathbb Z.$

Examples

• $15\in\mathbb Z$ is a multiple of $-5,$ since $15=(-3)\cdot(-5),$ and $-3\in\mathbb Z.$
• $\frac 12 x^2+\frac 1{12} x-1\in\mathbb Q[X]$ is a multiple of $\frac 34{x}-1$ since $\frac 12x^2+\frac 1{12} x-1=\left(\frac 23{x}+1\right)\left(\frac 34{x}-1\right),$ and $\frac 23{x}+1\in\mathbb Q[X].$

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

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References

Bibliography

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