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.

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