A ring $R$ (with 1) has IBN property if free $R$-modules have unique rank (e.g., commutative rings). In the same fashion, lets call $R$ a **good** ring if in every free $R$-module any independent set can be extended to a basis (not every commutative ring has this property). I have two questions:

Is the class of

**good**rings well-known?Is it true that every Noetherian ring has IBN property ?

Lectures on modules and rings, Prop. 1.8 and Prop. 1.13) . But how is this connected to your class of good rings? – Martin Brandenburg Aug 27 '13 at 10:01