I Primitive Elements and Free Modules ... ...

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I am reading Paul E. Bland's book, "Rings and Their Modules".

I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help in order to fully understand the proof of Proposition 4.3.14 ... ...

Proposition 4.3.14 reads as follows:


?temp_hash=5442b546862ff55ca769a0f78124e05a.png

?temp_hash=5442b546862ff55ca769a0f78124e05a.png



In the above proof by Bland we read the following:

" ... ... The induction hypothesis gives a basis ## \{ x, x'_2, \ ... \ ... \ x'_{n -1} \}## of ##M## and it follows that ## \{ x, x'_2, \ ... \ ... \ x'_{n - 1}, x'_n \}## is a basis of ##F## that contains ##x## ... ... "


My question is as follows:

Why/how exactly does it follow that ##\{ x, x'_2, \ ... \ ... \ x'_{n - 1}, x'_n \}## is a basis of ##F## that contains ##x##. ... ...


Help will be appreciated ...

Peter


====================================================================================================

It may help PFmembers reading this post to have access to Bland's definition of 'primitive element of a module' ... especially as it seems to me that the definition is a bit unusual ... so I am providing the same as follows:


?temp_hash=5442b546862ff55ca769a0f78124e05a.png



Hope that helps ... ...

Peter
 

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andrewkirk

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That ##\{x_1,...,x_n\}## is a basis for ##F## means the same as that
$$F=x_1R\oplus...\oplus x_nR$$
and since ##M\triangleq x_1R\oplus...\oplus x_{n-1}R## we have ##F=M\oplus x_nR##.

Since the induction hypothesis gives us that ##\{x, x'_2,...,x'_{n-1}\}## is a basis of ##M## we have
$$M = xR\oplus x'_2R\oplus x'_3R\oplus...\oplus x'_{n-1}$$
So we have
$$F=M\oplus x_nR
= xR\oplus x'_2R\oplus x'_3R\oplus...\oplus x'_{n-1}\oplus x_nR$$
which is equivalent to saying that ##\{x,x'_2,...,x'_{n-1},x_n\}## is a basis for ##F##.
 

Math Amateur

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Thanks Andrew ...

Appreciate your help ...

Peter
 

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