I am reading Dummit and Foote Sections 9.3 Polynomial Rings that are UFDs.
I have a problem understanding what D&F say regarding GCDs on page 306 at the end of Section 9.3 (see attached)
"we saw earlier that if R is a Unique Factorization Domain with field of fractions F and , then we can factor out the greatest common divisor d of the coefficients of p(x) to obtain p(x) = dp'(x) where p'(x) is irreducible in both R[x] and F[x]. Suppose now that R is an arbitrary integral domain with field of fractions F. In R the notion of greatest common divisor may not make sense, however, one might still ask if, say, a monic polynomial which is irreducible in R[x] is still irreducible in F[x] (i.e. whether the last statement in Corollary 6 is true). ... ...
My question is as follows: Why do D&F say "Suppose now that R is an arbitrary integral domain with field of fractions F. In R the notion of greatest common divisor may not make sense"? D&F's definition of GCD on page 274 (see attached) gives the definition for a GCD of two ring elements a and b for any commutative ring - and there are no conditions on the existence of the GCD - so why for an integral domain would we have a situation where the GCD does not make sense???
Can anyone clarify this for me?