Q as Module over Z: Proving Unfinite Generating Ability

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In summary: Not so much, no. I've just heard of it. But I'll look into it.the point is a finitely generated module iover Q has the same number of elements as Q, while that is less than the number of elements of R.I suggest you learn some basic set theory before going too deep into algebra.
  • #1
sab47
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Hey guys,

I'm self-teaching maths to preper myself for the next term of uni, so I'm reading this book on abstract algebra, and somewhere it says that R (the set of real numbers) is not finitely generated as a module over Q (set of rational numbers). Now, I can see that it's not, but i can't think of a rigorous proof for it. I thought maybe i hould just find a countr example like i did in a different case (Q is not finitely ggenerated over Z) but I'm prety bad at these counter examples! lol. Can anyone help me make sense of this? cause i prefer to understand everything before i continue to the next part.
 
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  • #2
What would it mean to find a counterexample here?
 
  • #3
DeadWolfe said:
What would it mean to find a counterexample here?

there was a similar statement which said Q is not finitely generated over Z. So what I did with that was i said if we assume q1,...,qn generate Q. Then take a z in Z which is coprime with the denominator of all members of the generatind set (i.e coprime with all qi) then 1/z cannot be generated by this set q1,...,qn . so Q is not finitely generated.

So this is what I meant by a counter example, finding something like 1/z above, which can't be generated by the generating set. Pehaps counter example isn't the best way to put it!
 
  • #4
for R over Q, merely the number of elements suffices.
 
  • #5
mathwonk said:
for R over Q, merely the number of elements suffices.

number of elements? how do you mean?
 
  • #6
How many elements can a finitely-generated module over Q possibly have? How many elements does R have?
 
  • #7
AKG said:
How many elements can a finitely-generated module over Q possibly have? How many elements does R have?

Should I somehow show that any finitely generated set over Q has finite number of elements? Sorry to be so slow, like I said I'm self teaching these things. There must be a theorem or something about number of elements of finitely generated modules which I've forgotten!
 
  • #8
sab47 said:
Should I somehow show that any finitely generated set over Q has finite number of elements?

No, of course not. Q is finitely generated over itself, how many elements does it have? What if F is a finitely generated free module over Q? How many elements does it have? Can this help us solve the more general problem?
 
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  • #9
do you know about countable, uncountable? this theory was introduced by cantor some 100 years ago.
 
  • #10
mathwonk said:
do you know about countable, uncountable? this theory was introduced by cantor some 100 years ago.

Not so much, no. I've just heard of it. But I'll look into it.
 
  • #11
the point is a finitely generated module iover Q has the same number of elements as Q, while that is less than the number of elements of R.
 
  • #12
I suggest you learn some basic set theory before going too deep into algebra. Knowledge of cardinalities, the schroder-bernstein theorem and Zorn's lemma are all pretty important prerequisites for studying modules and rings properly.
 

1. What is the concept of "Q as Module over Z"?

"Q as Module over Z" refers to the mathematical structure of the rational numbers (Q) being viewed as a module over the integers (Z). This means that we can use the operations of addition and scalar multiplication to manipulate the rational numbers, similar to how we would manipulate vectors in a vector space.

2. What does it mean to prove the "unfinite" generating ability of Q as a module over Z?

To prove the "unfinite" generating ability of Q as a module over Z means to show that every rational number can be written as a linear combination (using scalar multiplication and addition) of a finite number of elements in Z. In other words, we can use a finite number of integers to generate all of the rational numbers.

3. Why is it important to prove the unfinite generating ability of Q as a module over Z?

This proof is significant because it demonstrates the fundamental structure of the rational numbers and their relationship to the integers. It also allows us to better understand the behavior of rational numbers and their properties, which can be applied in various fields of mathematics.

4. How is the proof of unfinite generating ability of Q as a module over Z carried out?

The proof involves showing that any rational number can be expressed as a finite sum of fractions with denominators that are powers of a single prime number. This can be achieved using the fundamental theorem of arithmetic and properties of modular arithmetic.

5. Are there any real-world applications of this concept?

Yes, the concept of "Q as Module over Z" has applications in various fields such as cryptography, coding theory, and number theory. It also has implications in computer science, particularly in the design and analysis of algorithms for efficient computation with rational numbers.

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