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## Homework Statement

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

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:

Discussion of relevant definitions are given in my attempt at a solution ...

## Homework Equations

## The Attempt at a Solution

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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 ##M## generating a module ##N##" ...

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

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

Now if spanning ##M## is the same as generating ##M## ... then in Problem 1 we could take ## \mathscr{S} = \{ M \} ## as generating ##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 ##M## is said to be generated by a set ##\{ M_\alpha \}_\Delta ## of R-modules if there is an epimorphism ##\bigoplus_\Delta M_\alpha \rightarrow M##. ... ... "

Maybe this is the definition to use in Problem 1 above ... with ##\{ 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

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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 ...

Hope that helps ...

Peter

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