Q as a module over Z

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

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

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

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

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

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.