I am reading T. S. Blyth's book "Module Theory: An Approach to Linear Algebra" ... ... and am currently focussed on Chapter 2: Submodules; Intersections and Sums ... ...(adsbygoogle = window.adsbygoogle || []).push({});

I need help with understanding two claims that Blyth makes concerning submodules ...

The relevant text is as follows: ( see end of post for other text that may be relevant)

I have two questions concerning the above text ... ...

Question 1

In the above text we read:

" ... ... We know that ##A + B## is the smallest submodule of ##M## that contains both ##A## and ##B##, ... ... "

My question is: how exactly do we know this ... ? How would we formally and rigorously prove this ... ?

Question 2

In the above text we read:

" ... ... and that ##A \cap B## is the largest submodule contained in both ##A## and ##B##, ... ... "

My question is: how exactly do we know this ... ? How would we formally and rigorously prove this ... ?

Hope that someone can help with the above two questions ...

Peter

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PS Just in case readers need to reference some of Blyth's definitions or theorems in Chapter 2, I am providing the relevant text as follows:

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# I Submodules A + B and A intersect B ... Blyth Ch. 2

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