- #1

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## Homework Statement

I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.2 Free Modules ... ...

I need someone to check my solution to the first part of Problem 1(a) of Problem Set 2.2 ...

Problem 1(a) of Problem Set 2.2 reads as follows:

## Homework Equations

## The Attempt at a Solution

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My solution/proof of the first part of Problem 1(a) is as follows:

We claim that ##M \bigoplus N## is finitely generated ...Now ...

##M \bigoplus N =## the direct product ##M \times N## since we are dealing with the external direct sum of a finite number of modules ...

##M## finitely generated ##\Longrightarrow \exists## a finite subset ##X \subseteq M## such that

##M = \sum_X x_i R = \{x_1 r_1 + \ ... \ ... \ x_m r_m \mid x_i \in X, r_i \in R \}## ... ... ... ... (1)

##N## finitely generated ##\Longrightarrow \exists## a finite subset ##Y \subseteq N## such that

##N = \sum_Y y_i R = \{y_1 r_1 + \ ... \ ... \ y_n r_n \mid y_i \in Y, r_i \in R \}## ... ... ... ... (2)

##M \bigoplus N## finitely generated ##\Longrightarrow \exists## a finite subset ##S \subseteq M \bigoplus N## such that

##M \bigoplus N = \sum_S ( x_i, y_i ) R = \{ (x_1, y_1) r_1 + \ ... \ ... \ ( x_s, y_s) r_s \mid (x_i, y_i) \in S , r_i \in R \} ####= \{ (x_1 r_1, y_1 r_1) + \ ... \ ... \ + ( x_s r_s, y_s r_s) \}##

##= \{ (x_1 r_1 + \ ... \ ... \ + x_s r_s , y_1 r_1 + \ ... \ ... \ + y_s r_s \} ## ... ... ... ... ... (3)

Now if we take ##s \ge m, n## in (3) ... ...

Then the sum ##x_1 r_1 + \ ... \ ... \ + x_s r_s## ranging over all ##x_i## and ##r_i ## will generate all the elements in ##M## as the first variable in ##M \bigoplus N ##

... and the sum ##y_1 r_1 + \ ... \ ... \ + y_s r_s ## ranging over all ##y_i## and ##r_i## will generate all the elements in ##N## as the second variable in ##M \bigoplus N##

Since s is finite ... ##M \bigoplus N## is finitely generated ...

Can someone please critique my proof and either confirm it to be correct and/or point out the errors and shortcomings ...

Peter