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We know that (left) modules "are not as nice as vector spaces" because lots of properties of these ones don't fit to the most general modules. But il we restrict to modules on IBN ring (i.e. modules in which rank is well-defined), are we "at home" as regards to vector spaces properties such as:

1) all spanning sets have cardinality at least cardinality of basis (= rank of the module)?

2) all linearly independant sets have cardinality at most cardinality of basis (= rank of the module)?

3) all spaning sets contain a base?

4) all linearly independant sets can be completed to become a base?

5) all submodules have complements?

etc.

Or is the final answer simply: "NO !" ;-)?

(References in math literature welcome :-))

EDIT (02/07/15) : I browsed the book of Lam, Lectures on modules and rings, and have the feeling that modules over commutative ring with infinite basis behave exactly like vector spaces about questions asked here (because Lam deals essentialy with finite basis and modules like $R^n$...). Can someone tell me if this is right ?

Carré rond
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1 Answers1

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A ring satisfying 1) is said to have the "rank condition," and a ring satisfying 2) is said to satisfy the "strong rank condition." (Actually there is a right-left sidedness to the SRC which I gloss over for now.)

You can read about both of these conditions in the first chapter of Lam's Lectures on modules and rings. In general, the strong rank condition implies the rank condition, which in turn implies IBN.

I do not have answers off the top of my head for 3) and 4), but I feel like they've been discussed once before here.

It's also well-known that 5) is equivalent to $R$ being a semisimple ring.

rschwieb
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