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I am reading J A Beachy's Book, Introductory Lectures on Rings and Modules"... ...
I am currently focused on Chapter 2: Modules ... and in particular Section 2.4: Chain Conditions ...
I need help with the proof of Proposition 2.4.5 ...
Proposition 2.4.5 reads as follows:
In the above text by Beachy ... in the proof of part (a) ... we read the following:
"... ... ... Conversely, assume that ##N## and ##M/N## are Noetherian, and let ##M_0## be a submodule of ##M##. Then ##M_0 \cap N## and ##M_0 / ( M_0 \cap N ) \cong (M_0 + N) / N## are both finitely generated, so ##M_0## is finitely generated ... ... ...
I am very unsure of this part of the proof ... but overall Beachy seems to be trying to prove that an arbitrary submodule of ##M##, namely ##M_0##, is finitely generated ... ... and this means that ##M## is Noetherian ...
(Beachy, in his Proposition 2.4.3 has shown that every submodule of ##M## being finitely generated is equivalent to ##M## being Noetherian ... ... )
BUT ... I do not see how it follows in the above that ... ... ##M_0 \cap N## and ##M_0 / ( M_0 \cap N ) \cong (M_0 + N) / N## are both finitely generated ... ... AND ... exactly why it then follows that ##M_0## is finitely generated ... ...
Hope someone can help ...
Peter
I am currently focused on Chapter 2: Modules ... and in particular Section 2.4: Chain Conditions ...
I need help with the proof of Proposition 2.4.5 ...
Proposition 2.4.5 reads as follows:
In the above text by Beachy ... in the proof of part (a) ... we read the following:
"... ... ... Conversely, assume that ##N## and ##M/N## are Noetherian, and let ##M_0## be a submodule of ##M##. Then ##M_0 \cap N## and ##M_0 / ( M_0 \cap N ) \cong (M_0 + N) / N## are both finitely generated, so ##M_0## is finitely generated ... ... ...
I am very unsure of this part of the proof ... but overall Beachy seems to be trying to prove that an arbitrary submodule of ##M##, namely ##M_0##, is finitely generated ... ... and this means that ##M## is Noetherian ...
(Beachy, in his Proposition 2.4.3 has shown that every submodule of ##M## being finitely generated is equivalent to ##M## being Noetherian ... ... )
BUT ... I do not see how it follows in the above that ... ... ##M_0 \cap N## and ##M_0 / ( M_0 \cap N ) \cong (M_0 + N) / N## are both finitely generated ... ... AND ... exactly why it then follows that ##M_0## is finitely generated ... ...
Hope someone can help ...
Peter
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