then by (1.2.2) part (iv), PРis not disjoint from S = R \ P , which is impossible. Therefore S−1P is a proper idealРcontaining every maximal ideal, so it must be the unique maximal ideal. ♣Р1.2.10 RemarkР −1РIt is convenient to write the ideal S I as IRP . There is no ambiguity, because theРproduct of an element of I and an arbitrary element of R belongs to I.Р1.2.11 Localization of ModulesРIf M is an R-module and S a multiplicative subset of R, we can essentially repeat theРconstruction of (1.2.1) to form the localization of M by S, and thereby divide elementsРof M by elements of S.Ifx, y ∈ M and s, t ∈ S, we call (x, s) and (y, t) equivalent if forРsome u ∈ S, we have u(tx − sy) = 0. The equivalence class of (x, s) is denoted by x/s,Рand addition is defined byР x y tx + syР + = .Р s t st
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