Let R be a commutative ring and let P be a prime ideal in R. Let S be the elements of R that are not in P, which is sometimes written R-P. Since P is prime, S is closed under multiplication, so the fractions of R by S, written R/S, are well defined. In this case we have other words and symbols to describe the fraction ring of R by S.
This leads to an immediate and troubling ambiguity. Consider for a moment the ring of integers, which is often denoted Z. The prime number p generates a prime ideal, which is (unfortunately) indicated by the same letter p. Now Z localized about p, or Zp, is the ring of fractions without p in the denominator. Does this look familiar? Zp is the ring of integers mod p, a completely different animal. And to top things off, Zp is also standard notation for the p-adic numbers. good grief!
Part of our problem is the use of p for the prime number and the ideal generated by that number. Some have suggested the use of {p} for the ideal generated by p. Then the localization about p becomes Z{p}. That helps, in the context of the integers, but P is often a prime ideal in an arbitrary ring, and in that context we simply write Rp, so seeing Z{p} could be a bit comfusing. And this doesn't address the collision between the p-adic numbers and the cyclic group of order p.
I am not the first to confront this issue, and others have suggested the use of Z/pZ for the integers mod p. After all, when H is an ideal in a ring R, R/H is the homomorphic image. So Z/p or Z/pZ is naturally the homomorphic image of the integers mod p. This makes a great deal of sense.
Having said that, I and now going to paste a frown on my face and tell you that I'm not going to use this (clearer) notation, at least not today. Text books and professors all over the world use Zp for the integers mod p, and I just don't feel like bucking the system. I think it would confuse more than it helps. So the price I am willing to pay, for my conformist tendencies, is a rather nasty ambiguity between the integers mod p, the ring of fractions without p in the denominator, and the p-adic numbers. I hope the context, along with my occasional explanations, will make the notation clear.
Ok, where were we?
The notation RP indicates the ring of fractions with numerators in R and denominators in S = R-P. If H is an ideal in R, then HP is the localization of H at P, i.e. the fractions with numerators in H and denominators in S.
Any proper ideal in RP is the image of an ideal disjoint from S, in other words, an ideal wholly contained in P. Thus all proper ideals in RP are contained in the unique maximal ideal PP.
If a commutative ring has one maximal ideal it is called a local ring. As shown above, RP is a local ring, and PP is its unique maximal ideal. Examples of local rings include the rationals withh odd denominators (the localization of Z about 2), or quotients of integer polynomials without any factors of x2+1 in the denominator.
In a local ring R, all the nonunits are sequestered in the maximal ideal M. If a nonunit x is in R-M then x is contained in a maximal ideal other than M, which is a contradiction.