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

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In summary, Peter provided a summary of the content of the conversation. He said that he needs someone to check his solution to the first part of Problem 1(a) of Problem Set 2.2, and that Homework Equations is an easy way to proof that M is finitely generated.
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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:
Bland - Problem 1 - Problem Set 2.2 ... .....png
Can someone please critique my proof and either confirm it to be correct and/or point out the errors and shortcomings ...

Homework Equations

The Attempt at a Solution



[/B]
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
 

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  • #2
The easiest way to prove this is to construct a finite generating set. A natural candidate is
$$S\triangleq X^*\cup Y^* $$
where ##X^*\triangleq X\times\{0_N\}## and ##Y^*\triangleq \{0_M\}\times Y##
Choose an arbitrary element ##(a,b)\in M\oplus N## and use the fact that there exist ##r_1,...,r_m\in R## and ##s_1,...,s_n\in R## such that
##a=\sum_{j=1}^m r_jx_j## and ##b=\sum_{j=1}^n s_jy_j## to express ##(a,b)## as a finite sum of elements of ##S##.
 
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  • #3
andrewkirk said:
The easiest way to prove this is to construct a finite generating set. A natural candidate is
$$S\triangleq X^*\cup Y^* $$
where ##X^*\triangleq X\times\{0_N\}## and ##Y^*\triangleq \{0_M\}\times Y##
Choose an arbitrary element ##(a,b)\in M\oplus N## and use the fact that there exist ##r_1,...,r_m\in R## and ##s_1,...,s_n\in R## such that
##a=\sum_{j=1}^m r_jx_j## and ##b=\sum_{j=1}^n s_jy_j## to express ##(a,b)## as a finite sum of elements of ##S##.
Thanks Andrew ... indeed a much nicer way to do the proof ...

Peter
 

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

1. What is a finitely generated module?

A finitely generated module is a mathematical structure that consists of a set of elements and operations on those elements. It is generated by a finite number of elements, meaning that any element in the module can be expressed as a linear combination of those generators.

2. What is the Bland Problem 1(a) for finitely generated modules?

Bland Problem 1(a) is a problem in the field of commutative algebra that asks for a characterization of finitely generated modules over a commutative ring R that are flat and faithful. In other words, it seeks to determine the conditions under which a finitely generated module is flat and faithful.

3. What is Set 2.2 in relation to Bland Problem 1(a)?

Set 2.2 is a specific set of conditions that are proposed as a solution to Bland Problem 1(a). It consists of a set of necessary and sufficient conditions for a finitely generated module over a commutative ring R to be flat and faithful.

4. How is Bland Problem 1(a) related to other problems in commutative algebra?

Bland Problem 1(a) is part of a larger area of study in commutative algebra known as flatness and faithful flatness. It is closely related to other problems, such as the faithful flatness conjecture and Serre's Conjecture II, which also seek to characterize flat and faithful modules over commutative rings.

5. What are some current research directions in the study of Bland Problem 1(a) for finitely generated modules?

Some current research directions include investigating the role of torsion in Bland Problem 1(a), exploring connections to other areas of mathematics such as homological algebra and algebraic geometry, and studying the implications of Bland Problem 1(a) for other problems in commutative algebra.

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