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 ?