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 , so for example, with , we get

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.