Product of fractional ideals

For fractional ideals $I,J$ of an integral domain $R$ with fraction field $K$ , their product is

IJ=\left\{\sum_{\nu=1}^m x_\nu y_\nu: m\ge0,\ x_\nu\in I,\ y_\nu\in J\right\}\subseteq K.

The empty sum is zero. Finite sums are essential: the set of individual products alone need not be an additive subgroup.

Closure and algebraic laws

If $dI\subseteq R$ and $eJ\subseteq R$ , then $deIJ\subseteq R$ . Nonzero elements of $I$ and $J$ have a nonzero product, so $IJ$ is again a nonzero fractional ideal.

Multiplication is associative and commutative, with identity $R$ . On ordinary ideals it agrees with ideal multiplication. Invertible fractional ideals are precisely the units of this commutative monoid.

A calculation

Over $\mathbb Z$ ,

\left(\frac23\mathbb Z\right)\left(\frac57\mathbb Z\right) =\frac{10}{21}\mathbb Z.

This product is not set intersection or set union.

References

Distributivity proves closure under addition and scalar multiplication. Expanding finite sums proves associativity, commutativity, and the identity law.