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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:

View attachment 8067

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

We claim that \(\displaystyle M \bigoplus N\) is finitely generated ... Now ...

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

\(\displaystyle M\) finitely generated \(\displaystyle \Longrightarrow \exists\) a finite subset \(\displaystyle X \subseteq M\) such that

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

\(\displaystyle N\) finitely generated \(\displaystyle \Longrightarrow \exists\) a finite subset \(\displaystyle Y \subseteq N\) such that

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

\(\displaystyle M \bigoplus N\) finitely generated \(\displaystyle \Longrightarrow \exists\) a finite subset \(\displaystyle S \subseteq M \bigoplus N\) such that

\(\displaystyle M \bigoplus N = \sum_S ( x_i, y_i ) R = \{ (x_1, y_1) r_1 + \ ... \ ... \ ( x_s, y_s) r_s \mid (x_i, y_i) \in S , r_i \in R \} \)\(\displaystyle = \{ (x_1 r_1, y_1 r_1) + \ ... \ ... \ + ( x_s r_s, y_s r_s) \} \)\(\displaystyle = \{ (x_1 r_1 + \ ... \ ... \ + x_s r_s , y_1 r_1 + \ ... \ ... \ + y_s r_s \}\) ... ... ... ... ... (3)

Now if we take \(\displaystyle s \ge m, n \) in (3) ... ...Then the sum \(\displaystyle x_1 r_1 + \ ... \ ... \ + x_s r_s\) ranging over all \(\displaystyle x_i\) and \(\displaystyle r_i\) will generate all the elements in \(\displaystyle M\) as the first variable in \(\displaystyle M \bigoplus N \)

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

Since \(\displaystyle s\) is finite ... \(\displaystyle M \bigoplus N \) 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