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Basic Properties 8.1
Let M be an R-module.

(b) Every linearly independent subset S of M is contained in a maximal linearly
independent subset of M.
(c) M has a maximal linearly independent subset.
(d) If M is a free R-module, then every basis for M is a maximal linearly indepen
dent subset of M.
PROOF: 略

P217
Proposition 8.2
Let D be a division ring. Then the following statements are equivalent for a subset
B of a D-module M:
(a) B is a basis for M.
(b) B is a maximal linearly independent subset of M. Since every module has a
maximal linearly independent subset, every module over a division ring D has
a basis and is therefore a free D-module.
PROOF: Because the reader has already shown that every basis of a module is
a maximal linearly independent subset of the module, we only have to show that
(b) implies (a).
略
P218
This finishes the proof that a maximal linearly independent subset B of a
D-module M is a basis for M because it generates M.

Having established that all modules over division rings are free, we will have
a complete description of all nonzero rings R with the property that all R -modules
are free if we show that any nonzero ring with this property must be a division
ring. Because we are trying to describe when a ring is a division ring in terms of its
module theory, it is reasonable to expect that a module-theoretic description of
when a ring is a division ring would be helpful. We do this now in terms of the
properties of the R-module R.

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