In GCD domain every invertible ideal is principal
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This is the exercise in the book Commutative Rings by Kaplansky. Prove that in a GCD domain every invertible ideal is principal. I'm looking for some hints. Edit 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 we can write $I=(a_1/b_1,...,a_n/b_n)R$ , where $a_i, b_i$ are elements in $R$ . Since $R$ 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 tha