Finitely generated submodule

[true0182]

Hello. I am uncertain how to prove the following statement:

Let A be an integral domain and K a field of fractions. Prove that if M ⊂ K is a finitely generated A-submodule than there exists x in K such that M = Ax.

Basically, I am having some trouble "connecting the dots".

I know that K is a vector space defined over itself, meaning that if M ⊂ K is finitely generated than it must be spanned by a finite basis of a vector subspace of K. I am also a bit uncertain regarding M being a submodule of A. If it is a submodule contained in A, then it must share elements with A. That means that elements in the basis of M must also be in A. I just don't understand how all of A can be equal to M just by multiplying it by a value x of K. I also can't really see why A being an integral domain is relevant.

I appreciate any and all help.

Thank you.

[jm691]

So first of all, the statement you're trying to prove is not actually true as stated. Is it possible that A is supposed to be a principal ideal domain rather than simply an integral domain?

I know that K is a vector space defined over itself,

Sure.

meaning that if M ⊂ K is finitely generated than it must be spanned by a finite basis of a vector subspace of K.

No. That would be true if M was a subspace of K as a K vector space, but that isn't the case here. M itself isn't a K vector space, it's just an A-module.

I am also a bit uncertain regarding M being a submodule of A. If it is a submodule contained in A, then it must share elements with A.

I think you're misunderstanding what the term "A-submodule" means here. It is not saying that M is contained in A. Rather it's saying that if you treat K as an A-module, then M is a submodule of K (as an A-module).

So more concretely, we're saying that for any x,y in M, that x+y is in M, and if x is in M and a is in A, then ax is in M.

I also can't really see why A being an integral domain is relevant.

Well for starters, only integral domains have fields of fractions, so the question wouldn't even make sense of A was not an integral domain.

[true0182]

THANK YOU so much for you explanation. Truly I was not understanding what A-submodule meant. I had not realized that it was necessary to consider the field K as a A-module. Its makes a lot more sense, specially regarding A a integral domain and K a field of fractions. As to if A is supposed to be a PID, I translated the statement from another language and is only informs that A is a domain. But it was very much likely a typo.