Let us define free module over a ring (possibly without unity) as:
Def: M is said to be free module over ring R (possibly without unity) if there exist X subset of M such that X is LI and spans M. Any such X is called basis of M.
Def: If M free module over a ring R. Cardinality of any basis is called rank of M (we are allowing M to have two different ranks)
Def: R is said to have IBN property if any free module over R has fixed rank.
Question: Does following rings have IBN property
- Commutative ring
- Commutative ring R with property that there exist r in R such that rs never 0 for all s≠0
- ID without unity
In other words can we find free module over R with two different ranks where R one of above rings?
Motivation for question: Any commutative ring with unity or ID has IBN property.
Remark: This definition of free module is not at all same to what is called free object in the category of modules
Are there notes of some professor who have taught this definition of free module?