Hello. I am working with the subring Z[p/q] of the ring of integers Q. Among other things, I want to show that Z[1/2], as a Z-module, cannot be generated by an element a + b/2 in Z[1/2]. Otherwise, if we take the element 1/2 in Z[1/2] then for some m in Z we can write
1/2 = ma +mb/2.
So this implies a = 0 and b = 1 so Z[1/2] is generated by 1/2. But then, for 1 + 1/2 in Z[1/2],
1 + 1/2 = m/2
for some m in Z is impossible. This is where I am starting to get confused. Couldn't we just add the left hand side of the last equation and then say m had to of been 3?