gcd domain (original) (raw)

For example, 0 is a gcd of 0 and 0 in any D. In fact, if d is a gcd of 0 and 0, then d∣0. But 0∣0, so that 0∣d, which means that, for some x∈D, d=0⁢x=0. As a result, 0 is the unique gcd of 0 and 0.

In general, however, a gcd of two elements is not unique. For example, in the ring of integers, 1 and -1 are both gcd’s of two relatively prime elements. By definition, any two gcd’s of a pair of elements in D are associates of each other. Since the binary relation “being associates” of one anther is an equivalence relation (not a congruence relation!), we may define the gcd of a and b as the set

For example, as we have seen, GCD⁡(0,0)={0}. Also, for any a∈D, GCD⁡(a,1)=U⁡(D), the group of units of D.

If there is no confusion, let us denote gcd⁡(a,b) to be any element of GCD⁡(a,b).

If GCD⁡(a,b) contains a unit, then a and b are said to be relatively prime. If a is irreducible, then for any b∈D, a,b are either relatively prime, or a∣b.

An integral domain D is called a gcd domain if any two elements of D, not both zero, have a gcd. In other words, D is a gcd domain if for any a,b∈D, GCD⁡(a,b)≠∅.

Remarks

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  A Bezout domain is always a gcd domain. A gcd domain D is a Bezout domain if gcd⁡(a,b)=r⁢a+s⁢b for any a,b∈D and some r,s∈D.
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  Given an integral domain, one can similarly define an lcm of two elements a,b: it is an element c such that a∣c and b∣c, and if d is an element such that a∣d and b∣d, then c∣d. Then, a_lcm domain_ is an integral domain such that every pair of elements has a lcm. As it turns out, the two notions are equivalent: an integral domain is lcm iff it is gcd.

The following diagram indicates how the different domains are related:

References

• 1 D. D. Anderson, Advances in Commutative Ring Theory: Extensions of Unique Factorization, A Survey, 3rd Edition, CRC Press (1999)
• 2 D. D. Anderson, Non-Noetherian Commutative Ring Theory: GCD Domains, Gauss’ Lemma, and Contents of Polynomials, Springer (2009)
• 3 D. D. Anderson (editor), Factorizations in Integral Domains, CRC Press (1997)