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