- #1

Math Amateur

Gold Member

MHB

- 3,998

- 48

Currently I am focused on Section 4.2 Noetherian and Artinian Modules ... ...

I need some help in order to make a meaningful start on Problem 1, Problem Set 4.1 ...

Problem 1, Problem Set 4.1 reads as follows:

View attachment 8097

Can someone please help me to make a meaningful start on Problem 1 above ... as well as clarifying the definition of one module generating another ... see my notes following ... ...

I am having a bit of trouble pinning down a definition from Bland that gives me the meaning of a module \(\displaystyle M\) generating a module \(\displaystyle N\) ...

Now ... on page 104 of Bland (see first page of Bland Section 4.1 below) we read the following:

" ... ... If \(\displaystyle \mathscr{S}\) is a set of submodules of \(\displaystyle M\) such that \(\displaystyle M = \sum_{ \mathscr{S} } N\), then \(\displaystyle \mathscr{S}\) is said to span \(\displaystyle M\) ... ... "

Now if spanning \(\displaystyle M\) is the same as generating \(\displaystyle M\) ... so in Problem 1 we could take \(\displaystyle \mathscr{S} = \{ M \} \) as generating \(\displaystyle N\) ... but is that the correct interpretation of "span" and would the definition be useful in Problem 1 ... ?

Alternatively ... on page 105 we read in Definition 4.1.2 (see Bland's text displayed below ... )

" ... ... An R-module \(\displaystyle M\) is said to be generated by a set \(\displaystyle \{ M_\alpha \}_\Delta \) of R-modules if there is an epimorphism \(\displaystyle \bigoplus_\Delta M_\alpha \rightarrow M\). ... ... "

Maybe this is the definition to se in Problem 1 above ... with \(\displaystyle \{ M_\alpha \}_\Delta = \{ N \}\) ... ... is that the correct start on the problem ...?

Can someone please clarify the above ... and then, further, help me to make a meaningful start on the problem ...Help will be appreciated,

Peter==========================================================================================In order to give readers the definitions, notation and context of Bland's treatment of generating and cogenerating classes, I am providing access to Bland's Section 4.1 ... as follows ...

https://www.physicsforums.com/attachments/8098

https://www.physicsforums.com/attachments/8099

View attachment 8100

Hope that helps ...

Peter