## Explain the difference between a basis and a generating set for an ideal in an algebraic integer ring

Let $I$ be an ideal in a ring $\mathcal{O}$ of algebraic integers in some algebraic number field $K$. Explain the difference between a basis for $I$ and a generating set for $I$.

Let us recall our definitions of ‘basis’ and ‘generating set’ in this context. We say $B$ is a basis for $I$ if every element of $I$ can be written uniquely as a $\mathbb{Z}$-linear combination of elements from $B$. We say $A$ is a generating set for $I$ if every element of $I$ is a (not necessarily unique) $\mathcal{O}$-linear combination of elements from $A$. In the language of modules, $A$ is merely a generating set for $I$ as an $\mathcal{O}$-module, while $B$ is a free generating set for $I$ as a $\mathbb{Z}$-module.