Why do we use *fractional* ideals in construction of the class group?

Question

I am trying to give a simplified explanation of the class group to students at the upper undergraduate level, who likely have a basic understanding of ring and group theory. While my motivation for constructing the class group as the quotient of the ideal group by the principal ideal group seems clear, I am struggling to justify the extension to fractional ideals. What necessitates this? In the example of $\mathbb{Z}[\sqrt{-5}]$, we can show that the product of any two non-principal ideals is principal, which holds with the class group being $\mathbb{Z}/2\mathbb{Z}$. What is the problem with regular ideals that necessitates the shift to fractional ideals?

Answer 1

Nothing necessitates the use of fractional ideals, you can define the ideal class group perfectly well without them, see https://math.stackexchange.com/a/296094/31917 . It is maybe just slightly neater / mathematically more convenient to quotient a group by an obvious subgroup than to use the equivalence relation mentioned there, but the definitions are equivalent.

Answer 2

An integral domain $A$ is a Dedekind ring if and only if the set of fractional ideals forms an abelian group under ideal multiplication. Since we want to assign the class number to any Dedekind ring, it seems very natural to consider the group of fractional ideals.