Principal fractional ideal

Let $R$ be an integral domain with fraction field $K$ . A principal fractional ideal is a fractional ideal of the form

aR=\{ar:r\in R\},\qquad a\in K^\times.

The generator need not lie in $R$ . The nonzero convention for fractional ideals is in force.

Examples and equality

For $R=\mathbb Z$ , both $2\mathbb Z$ and $\tfrac12\mathbb Z$ are principal fractional ideals; only the first is an ordinary ideal contained in $\mathbb Z$ .

One has $aR=bR$ exactly when $a/b$ is a unit of $R$ . Moreover $(aR)(bR)=abR$ , and $(aR)^{-1}=a^{-1}R$ . Thus principal fractional ideals form a subgroup of the invertible fractional ideals.

Class groups

Quotienting invertible fractional ideals by this subgroup gives the Picard group of the domain. For a ring of integers this recovers the ideal class group.

References

The common-denominator property follows by writing $a=c/d$ . Equality and the product formulas follow directly from the definition.