I am reading Paul E. Bland's book, "Rings and Their Modules".(adsbygoogle = window.adsbygoogle || []).push({});

I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.4 ... ...

Proposition 4.2.4 reads as follows:

I need help to fully understand Part of the proof proving that ##(2) \Longrightarrow (3)## ...

In that part of the proof Bland seems to be assuming that

## \bigcap_F M_\alpha = N ##

if and only if

## \bigcap_F (M_\alpha / N ) = 0 ##

In other words, if ##F = \{ 1, 2, 3 \}## then

##M_1 \cap M_2 \cap M_3##

if and only if

##M_1 / N \cap M_2 / N \cap M_3 / N##

But whyis this the case ... ...exactly

... ... how do we formally and rigorously demonstrate that this is true ...

Hope someone can help ...

Peter

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

Proposition 4.2.4 refers to the (possibly not well known) concept of cogeneration so I am providing Section 4.1 Generating as Cogenerating Classes ... ... as follows ...

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# I Artinian Modules - Bland - Proposition 4.24

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