Let be an ideal in a ring of algebraic integers in some algebraic number field . Explain the difference between a basis for and a generating set for .
Let us recall our definitions of ‘basis’ and ‘generating set’ in this context. We say is a basis for if every element of can be written uniquely as a -linear combination of elements from . We say is a generating set for if every element of is a (not necessarily unique) -linear combination of elements from . In the language of modules, is merely a generating set for as an -module, while is a free generating set for as a -module.