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: 


1. Is the class of **good** rings well-known?
2. Is it true that every Noetherian ring has IBN property ?



A class of rings related to rings with IBN property
The Rings you are calling good, are called (left or right) Steinitz. They are characterized as: **A ring is (left) right Steinitz if and only if it is (left) right perfect and local**. This means that every (left) right Steinitz ring has IBN property. You may like to see the following


**[1]** Chew, Neggers, On the extension of linearly independent subsets of free modules to bases, Proc. Amer. Math. Soc. 24(1970), 466-470. 


**[2]** Brodskii, Steinitz Modules, Mat. Issled. 7(1972), 14-28, 284. 


For (2) as Martin said you can see Lam's Book. 


