Why is R not finitely generated as a module over Q?

[sab47]

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.

 

[DeadWolfe]

What would it mean to find a counterexample here?

 

[sab47]

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!

 

[mathwonk]

for R over Q, merely the number of elements suffices.

 

[sab47]

for R over Q, merely the number of elements suffices.

number of elements? how do you mean?

 

[AKG]

How many elements can a finitely-generated module over Q possibly have? How many elements does R have?

 

[sab47]

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!

 

[DeadWolfe]

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?

 

[mathwonk]

do you know about countable, uncountable? this theory was introduced by cantor some 100 years ago.

 

[sab47]

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.

 

[mathwonk]

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.

 

[DeadWolfe]

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.