Ok so when we say
we are not talking about divisibility over Z, but rather over some other integral domain?
(On a side note you just answered a lingering question I had - I often see the term "localised at a prime", but did note know what it meant)
Sorry to confuse you further: depends on how you view things.
If you are thinking about
, then it is divisibility over
. However, this is not going to be very useful when you start doing more fun things because, for example, we want to construct the p-adic integers
and do things over there or other overrings.
So, how can we do this? This is probably the place to introduce discrete valuation rings (DVRs) and valuation rings. A discrete valuation ring R is a PID with a unique maximal ideal, so we can put a discrete valuation
with the properties:
A valuation is when you relax v to take value in a totally ordered group, and a valuation ring is when you relax the PID to just being an integral domain R with valuation v on K=Frac(R) such that
The maximal ideal
allows us to define v by
, and we extend to
(check this is well-defined!). Of course, from this we can also recover
--- note the we need the intersection with R here, otherwise you will end up with exactly where you are confused before --- because
a valuation ring.