The greatest common divisor of $a$ and $b$ is a linear combination of $a$ and $b$. In general, in what kind of ring does this hold?

Rings in which every two-generated ideal is principal $\rm\:(a,b) = (c)\:$ are called Bezout rings, since they are precisely the rings where gcds exist and have linear (Bezout) form. For suppose that $\rm\:(a,b) = (c).\:$ Then $\rm\:(c)\supseteq (a),(b)\:\Rightarrow\: c\mid a,b,\:$ so $\rm\:c\:$ is a common divisor of $\rm\:a,b.\:$ Conversely $\rm\:(a,b)\supseteq (c)\:\Rightarrow\: c = ja + k b\:$ so $\rm\:d\mid a,b\:\Rightarrow\:d\mid c,\:$ so $\rm\:c\:$ is a greatest common divisor (greatest in terms of divisibility order).

Bezout domains lie between PIDs and GCD domains in the following list of domains closely related to GCD domains.

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PID: $\ \ $ every ideal is principal

Bezout: $\ \ $ every ideal (a,b) is principal

GCD: $\ \ $ (x,y) := gcd(x,y) exists for all x,y

SCH: $\ \ $ Schreier = pre-Schreier & integrally closed

SCH0: $\ \ $ pre-Schreier: a|bc $\, \Rightarrow\, $ a = BC, B|b, C|c

D: $\ \ $ (a,b) = 1 & a|bc $\,\Rightarrow\,$ a|c

PP: $\ \ $ (a,b) = (a,c) = 1 $\,\Rightarrow\,$ (a,bc) = 1

GL: $\ \ $ Gauss Lemma: product of primitive polys is primitive

GL2: $\ \ $ Gauss Lemma holds for all polys of degree 1

AP: $\ \ $ atoms are prime [i.e. PP restricted to atomic a]

Since atomic & AP $\,\Rightarrow\,$ UFD, reversing the above UFD $\,\Rightarrow\,$ AP path shows that in atomic domains all these properties (except PID, Bezout) collapse, becoming all equivalent to UFD.

There are also many properties known equivalent to D, e.g.

[a] $\ \ $ (a,b) = 1 $\,\Rightarrow\,$ a|bc $\,\Rightarrow\,$ a|c

[b] $\ \ $ (a,b) = 1 $\,\Rightarrow\,$ a,b|c $\,\Rightarrow\,$ ab|c

[c] $\ \ $ (a,b) = 1 $\,\Rightarrow\,$ (a)/\(b) = (ab)

[d] $\ \ $ (a,b) exists $\,\Rightarrow\,$ lcm(a,b) exists

[e] $\ \ $ a + b X irreducible $\,\Rightarrow\,$ prime for b $\ne$ 0 (deg = 1)


I think you are looking for Bézout domain, a well known concept in ring theory, generalizing the notion of principal ideal rings.

Tags:

Ring Theory

Divisibility

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