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Greatest common divisor
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== In commutative rings == {{See also|Divisibility (ring theory)}} {{More citations needed section|date=August 2024}} The notion of greatest common divisor can more generally be defined for elements of an arbitrary [[commutative ring]], although in general there need not exist one for every pair of elements.<ref>{{cite book |last=Lovett |first=Stephen |date=2015 |title=Abstract Algebra: Structures and Applications |chapter=Divisibility in Commutative Rings |pages=267–318 |publisher=CRC Press |publication-place=Boca Raton |isbn=9781482248913}}</ref> * If {{mvar|R}} is a commutative ring, and {{mvar|a}} and {{mvar|b}} are in {{mvar|R}}, then an element {{mvar|d}} of {{mvar|R}} is called a ''common divisor'' of {{mvar|a}} and {{mvar|b}} if it divides both {{mvar|a}} and {{mvar|b}} (that is, if there are elements {{mvar|x}} and {{mvar|y}} in {{mvar|R}} such that ''d''·''x'' = ''a'' and ''d''·''y'' = ''b''). * If {{mvar|d}} is a common divisor of {{mvar|a}} and {{mvar|b}}, and every common divisor of {{mvar|a}} and {{mvar|b}} divides {{mvar|d}}, then {{mvar|d}} is called a ''greatest common divisor'' of {{mvar|a}} and ''b''. With this definition, two elements {{mvar|a}} and {{mvar|b}} may very well have several greatest common divisors, or none at all. If {{mvar|R}} is an [[integral domain]], then any two GCDs of {{mvar|a}} and {{mvar|b}} must be [[associate elements]], since by definition either one must divide the other. Indeed, if a GCD exists, any one of its associates is a GCD as well. Existence of a GCD is not assured in arbitrary integral domains. However, if {{mvar|R}} is a [[unique factorization domain]] or any other [[GCD domain]], then any two elements have a GCD. If {{mvar|R}} is a [[Euclidean domain]] in which euclidean division is given algorithmically (as is the case for instance when {{math|1=''R'' = ''F''[''X'']}} where {{mvar|F}} is a [[field (mathematics)|field]], or when {{mvar|R}} is the ring of [[Gaussian integer]]s), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure. The following is an example of an integral domain with two elements that do not have a GCD: : <math>R = \mathbb{Z}\left[\sqrt{-3}\,\,\right],\quad a = 4 = 2\cdot 2 = \left(1+\sqrt{-3}\,\,\right)\left(1-\sqrt{-3}\,\,\right),\quad b = \left(1+\sqrt{-3}\,\,\right)\cdot 2.</math> The elements {{math|2}} and {{math|1 + {{sqrt|−3}}}} are two [[maximal common divisor]]s (that is, any common divisor which is a multiple of {{math|2}} is associated to {{math|2}}, the same holds for {{math|1 + {{sqrt|−3}}}}, but they are not associated, so there is no greatest common divisor of {{mvar|a}} and {{math|''b''}}. Corresponding to the Bézout property we may, in any commutative ring, consider the collection of elements of the form {{math|''pa'' + ''qb''}}, where {{mvar|p}} and {{mvar|q}} range over the ring. This is the [[ideal (ring theory)|ideal]] generated by {{mvar|a}} and {{mvar|b}}, and is denoted simply {{math|(''a'', ''b'')}}. In a ring all of whose ideals are principal (a [[principal ideal domain]] or PID), this ideal will be identical with the set of multiples of some ring element {{math|''d''}}; then this {{mvar|d}} is a greatest common divisor of {{mvar|a}} and {{math|''b''}}. But the ideal {{math|(''a'', ''b'')}} can be useful even when there is no greatest common divisor of {{mvar|a}} and {{math|''b''}}. (Indeed, [[Ernst Kummer]] used this ideal as a replacement for a GCD in his treatment of [[Fermat's Last Theorem]], although he envisioned it as the set of multiples of some hypothetical, or ''ideal'', ring element {{mvar|d}}, whence the ring-theoretic term.)
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