In GCD domain every invertible ideal is principal

This is the exercise in the book Commutative Rings by Kaplansky.

Prove that in GCD-domain every invertible ideal is principal.

I’m looking for some hints.


After understanding the hint, here is my approach:

Let $I$ be an invertible ideal of a GCD-domain $R$. Because $I$ is finitely generated as $R$-module $I=(a_1/b_1,…,a_n/b_n)R$, where $a_i, b_i$ are elements in $R$.

Because $R$ is a GCD-domain we can choose $a_i, b_i$ such that $(a_i,b_i)=1$.

By hypothesis $R$ is also an LCM-domain. Let $c$ be the least common multiple of $b_i$’s, $d$ be the greatest common divisor of $a_i$’s. It is easy to see that $I^{-1}=(c/d)R$.

Because of invertibility there exist $m_i$’s of $I$ such that $m_1(c/d)+\cdots+m_n(c/d)=1$.

We conclude that $I=uR$, where $u=m_1+\cdots+m_n$, for if $x\in I$, $x=x\cdot1=xm_1(c/d)+\cdots+xm_n(c/d)=x(c/d)u$.

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