Algebraic Number Theory
《Algebraic Number Theory》 · 第 21 页 · Chapter I，Section 3「Ideals」，Proposition 3.8
The fractional ideals form an abelian group, the ideal group $J_K$ of $K$. The identity element is $(1)=\mathcal O$, and the inverse of $\mathfrak a$ is $\mathfrak a^{-1}=\{x\in K\mid x\mathfrak a\subseteq\mathcal O\}$.