localization (original) (raw)

Let R be a commutative ring and let S be a nonempty multiplicative subset of R. The localization of R at S is the ring S-1⁢R whose elements are equivalence classes of R×S under the equivalence relation (a,s)∼(b,t) if r⁢(a⁢t-b⁢s)=0 for some r∈S. Addition and multiplication in S-1⁢R are defined by:

• •
  (a,s)+(b,t)=(a⁢t+b⁢s,s⁢t)
• •
  (a,s)⋅(b,t)=(a⋅b,s⋅t)

The equivalence class of (a,s) in S-1⁢R is usually denoted a/s. For a∈R, the localization of R at the minimal multiplicative set containing a is written as Ra. When S is the complement of a prime ideal 𝔭 in R, the localization of R at S is written R𝔭.