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**Jumping** Thank you for the help Tonio. My goal is to show that the subring Z[1/2] of Q, when considered as a Z-module, cannot be finitely generated. I figure I ought to first show that Z[1/2] cannot be generated by a single element from Z[1/2].

I think my confusion starts when I was writing on my scratch paper that

Z[1/2] = {a + b*(1/2) : a, b are in Z}.

This, of course, is false. We can define $\displaystyle \displaystyle{\mathbb{Z}\left[\frac{1}{2}\right]:=\{f(1/2)\,;\,\,f(x)\in\mathbb{Z}[x]\}$ , so for example, with $\displaystyle f(x)=x^4$ , we get

$\displaystyle \displaystyle{f\left(\frac{1}{2}\right)=\frac{1}{1 6}\in\mathbb{Z}\left[\frac{1}{2}\right]-\left\{a+b\cdot\frac{1}{2}\,;\,\,a,b\in\mathbb{Z}\ right\}}$

But, Z[1/2] is a subring of Q and so it also has a multiplicative structure. For example 1/2 is in Z[1/2] and so (1/2)(1/2) = 1/4 is also in Z[1/2]. But I don't see how I can produce an element 1/4 in Z[1/2] by my above description of Z[1/2].

So I think what I am really trying to do is show that if a is any element of Z[1/2], then I can find some element b in Z[1/2] so that it is impossible to write

b = m*a

for any m in Z. This would then show that the Z-module Z[1/2] cannot be generated by any single element a in Z[1/2].

I will keep working on this. Thanks again Tonio.