Quotient of fractional ideals

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

(I:J)=\{x\in K:xJ\subseteq I\}.

This is a nonzero fractional ideal. The ambient field matters: restricting $x$ to $R$ would give the ring-theoretic colon ideal $(I:J)\cap R$ .

Why it is fractional

Choose $0\ne a\in I$ and $0\ne d\in R$ with $dJ\subseteq R$ . Then $ad\in(I:J)$ , so it is nonzero. Choose $0\ne b\in J$ ; since $b(I:J)\subseteq I$ , a denominator for $I$ , together with a denominator for $b$ , yields a common denominator for $(I:J)$ .

Inverses and multipliers

The candidate inverse of $I$ is $(R:I)$ . It is an actual multiplicative inverse exactly when $I$ is invertible. One must not cancel a noninvertible fractional ideal in a product equation.

The quotient $(I:I)$ is the multiplier ring of $I$ , which may be larger than $R$ .

References

The displayed containment and denominator argument verify the definition over arbitrary integral domains; no Noetherian hypothesis is used.