# Finitely Generated Modules - Bland Problem 1(a), Problem Set 2.2

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In summary, the conversation was about a problem set and someone asking for a critique of their proof for the first part of a problem. The proof involves showing that a direct sum of two finitely generated modules is also finitely generated. The summary provides a brief overview of the proof and suggests a different approach to proving the statement.
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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:
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

Peter said:
We claim that $$\displaystyle M \bigoplus N$$ is finitely generated ...

$$\displaystyle M \bigoplus N$$ = 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)
So each $x \in M$ can be written as a finite sum $x = \sum_X x_i r_i$ ($r_i \in R$).
So each $y \in N$ can be written as a finite sum $y = \sum_Y y_i s_i$ ($s_i \in R$).
Peter said:
$$\displaystyle M \bigoplus N$$ finitely generated $$\displaystyle \Longrightarrow \exists$$ a finite subset $$\displaystyle S \subseteq M \bigoplus N$$ such that

This must be: To prove that $M \bigoplus N$ is finitely generated, so we have to prove that there exists a finite subset $S \subseteq M \bigoplus N$ such that $M \bigoplus N = \sum_S z_i R$.

I think it has to be this way:
Define $X'=\{ (x_i, 0) | x_i \in X \}$ and $Y'=\{ (0, y_i) | y_i \in Y \}$ and $Z=X' \cup Y'$. Then $Z \subset M \bigoplus N$ and $Z$ is finite.

Take $z \in M \bigoplus N$, then $z=(m,n)$ with $m \in M$ and $n \in N$.
$m$ can be written as $m = \sum_X x_i r_i$ and $n$ can be written as $n = \sum_Y y_i s_i$.

Then $z = (m,n) = (\sum_X x_i r_i, \sum_Y y_i s_i) = (\sum_X x_i r_i, 0) + (0, \sum_Y y_i s_i) = \sum_X (x_i, 0)r_i + \sum_Y (0, y_i)s_i = \sum_{X'} x'_i r_i +\sum_{Y'} y'_i s_i = \sum_Z z_i t_i$

in which $x'_i=(x_i, 0) \in X'$ and $y'_i=(0, y_i) \in Y'$ and $t_i \in R$.

This proves that $M \bigoplus N$ is finitely generated.

## 1. What is a finitely generated module?

A finitely generated module is a module that can be generated by a finite number of elements. This means that every element in the module can be written as a linear combination of the finite set of generators.

## 2. What does it mean for a module to be Noetherian?

A Noetherian module is a module that satisfies the ascending chain condition. This means that every increasing sequence of submodules eventually stabilizes, i.e. there is a finite number of submodules in the sequence.

## 3. How is the Bland Problem 1(a) related to finitely generated modules?

Bland Problem 1(a) is a question about whether every submodule of a finitely generated module is finitely generated. It is related to finitely generated modules because it deals with the generation of submodules within a finite set of generators.

## 4. Can you give an example of a module that is not finitely generated?

Yes, the ring of polynomials with infinitely many variables is an example of a module that is not finitely generated. This is because it cannot be generated by a finite number of elements, as there are infinitely many variables in the ring.

## 5. How is Problem Set 2.2 different from Bland Problem 1(a)?

Problem Set 2.2 is a collection of exercises related to finitely generated modules, while Bland Problem 1(a) is a specific question about submodules of finitely generated modules. Problem Set 2.2 may include other topics and concepts related to finitely generated modules, while Bland Problem 1(a) is solely focused on the generation of submodules.

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