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I am reading Paul E. Bland's book, "Rings and Their Modules".
I am trying to understand Section 2.2 on free modules and need help with the proof of Proposition 2.2.11.
Proposition 2.2.11 and its proof read as follows:View attachment 3588
Proposition 2.2.11 relies on the definition of an IBN-ring so I am providing Bland's definition of an IBN-ring which reads as follows:
View attachment 3589
Now in the proof of Proposition 2.2.11, Bland defines $$R$$ as a commutative ring ... ... and then has to show (see above definition of an IBN-ring) that for every free $$R$$-module $$F$$, any two bases of $$F$$ have the same cardinality.Bland then considers a free module $$F$$ ... ... ... ... BUT ... instead of considering bases of $$F$$ ... ... ... ... Bland, instead, shows that $$\{ x_\alpha + F \mathscr{m} \}_\Delta$$ is a basis for the vector space $$F/F \mathscr{m}$$ ... ... and also shows that $$\text{dim}_{R/ \mathscr{m}} ( F/F \mathscr{m} ) = \text{ card } ( \Delta )$$
... ... and then shows that
$$\text{ card } ( \Delta ) = \text{ card } ( \Gamma )$$
for any other basis $$\Gamma$$ of the vector space $$F/F \mathscr{m}$$
BUT ... ... we should be showing that all bases of $$F$$ (and NOT $$F/F \mathscr{m}$$ ) have the same cardinality ? !... ... so then ... ... how has Bland shown that every free $$R$$-module $$F$$ of the ring $$R$$ has bases of the same cardinality?Could someone explain the logic of Bland's proof of Proposition 2.2.11 ...
I would really appreciate help ...
Peter
I am trying to understand Section 2.2 on free modules and need help with the proof of Proposition 2.2.11.
Proposition 2.2.11 and its proof read as follows:View attachment 3588
Proposition 2.2.11 relies on the definition of an IBN-ring so I am providing Bland's definition of an IBN-ring which reads as follows:
View attachment 3589
Now in the proof of Proposition 2.2.11, Bland defines $$R$$ as a commutative ring ... ... and then has to show (see above definition of an IBN-ring) that for every free $$R$$-module $$F$$, any two bases of $$F$$ have the same cardinality.Bland then considers a free module $$F$$ ... ... ... ... BUT ... instead of considering bases of $$F$$ ... ... ... ... Bland, instead, shows that $$\{ x_\alpha + F \mathscr{m} \}_\Delta$$ is a basis for the vector space $$F/F \mathscr{m}$$ ... ... and also shows that $$\text{dim}_{R/ \mathscr{m}} ( F/F \mathscr{m} ) = \text{ card } ( \Delta )$$
... ... and then shows that
$$\text{ card } ( \Delta ) = \text{ card } ( \Gamma )$$
for any other basis $$\Gamma$$ of the vector space $$F/F \mathscr{m}$$
BUT ... ... we should be showing that all bases of $$F$$ (and NOT $$F/F \mathscr{m}$$ ) have the same cardinality ? !... ... so then ... ... how has Bland shown that every free $$R$$-module $$F$$ of the ring $$R$$ has bases of the same cardinality?Could someone explain the logic of Bland's proof of Proposition 2.2.11 ...
I would really appreciate help ...
Peter
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