Non-integral domain for which there still exists a gcd for each pair of elements.

Question

Does there exists an non-integral domain for which we still have a gcd for each pair of elements (a,b)? Here, when I say gcd, I mean the definition of gcd for commutative rings given by wikipedia.

Answer 1

Any PIR (Prinicpal Ideal Ring) not a domain works, since $(a,b) = (c)\Rightarrow\, c\,$ is a gcd of $a,b$. One simple way to construct such a ring is to quotient a PID by a non-prime ideal, e.g. $\,\Bbb Z/4.\,$ For more exotic examples we can employ semigroup rings using the following.

THEOREM $\ \ $ TFAE for a semigroup ring R[S], with unitary ring R, and nonzero torsion-free cancellative monoid S.

1) $\ $ R[S] is a PIR (Principal Ideal Ring)
2) $\ $ R[S] is a general ZPI-ring (i.e. a Dedekind ring, see below)
3) $\ $ R[S] is a multiplication ring (i.e. $\rm\ I \supset\ J \Rightarrow\ I\ |\ J\ $ for ideals $\rm\:I,J\:$)
4) $\ $ R is a finite direct sum of fields, and S is isomorphic to $\mathbb Z$ or $\mathbb N$

A general ZPI-ring is a ring theoretic analog of a Dedekind domain i.e. a ring where every ideal is a finite product of prime ideals. A unitary ring R is a general ZPI-ring $\iff$ R is a finite direct sum of Dedekind domains and special primary rings (aka SPIR = special PIR) i.e. local PIRs with nilpotent max ideals. ZPI comes from the German phrase "Zerlegung in Primideale" = factorization in prime ideals. The classical results on Dedekind domains were extended to rings with zero divisors by S. Mori circa 1940, then later by K. Asano and, more recently, by R. Gilmer. See Gilmer's book "Commutative Semigroup Rings" sections 18 (and section 13 for the domain case).