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Post Closed as "Needs details or clarity" by David White, Andrés E. Caicedo, Ryan Budney, Todd Trimble, Eric Naslund
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user39121
user39121

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 knownwell-known?

  2. Is it true that every Noetherian ring has IBN property ?

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 known-known?

  2. Is it true that every Noetherian ring has IBN property ?

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 ?

Improved
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user39121
user39121

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 known-known?

  2. Is it true that every Noetherian ring has IBN property ?

A ring $R$ (with 1) has IBN property if free $R$-modules have unique rank. 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. I have two questions:

  1. Is the class of good rings known-known?

  2. Is it true that every Noetherian ring has IBN property ?

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 known-known?

  2. Is it true that every Noetherian ring has IBN property ?

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user39121
user39121

A class of rings related to rings with IBN property

A ring $R$ (with 1) has IBN property if free $R$-modules have unique rank. 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. I have two questions:

  1. Is the class of good rings known-known?

  2. Is it true that every Noetherian ring has IBN property ?