Abstract
If a, b ∈ R a commutative ring, we say that b divides a (denoted b|a) if there is an element c ∈ R: a = bc. One checks b|a ⇔ aR = (a) ⊆ (b) = bR. Two elements are associated (in divisibility) denoted by a ~ b if b|a and a|b. This is an equivalence relation and a ~ b ⇔ aR = bR An element p is called
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irreducible if it is not a unit and has no other divisors then units and associated elements;
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reducible if it is not irreducible;
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prime if it is not a unit and p|a.b ⇒ p|a or p|b;
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greatest common divisor for r 1, r 2,…, r n if p|r i (1 ≤ i ≤ n) and if d|r i(1 ≤ i ≤ n) then d|p;
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least common multiple is defined similarly;
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Each prime element is irreducible.
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© 1998 Springer Science+Business Media Dordrecht
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Cǎlugǎreanu, G., Hamburg, P. (1998). Divisibility in Integral Domains. In: Exercises in Basic Ring Theory. Kluwer Texts in the Mathematical Sciences, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9004-4_6
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DOI: https://doi.org/10.1007/978-94-015-9004-4_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4985-8
Online ISBN: 978-94-015-9004-4
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