V(I)= implies I=R

Note that most of the notation used here is defined in the entry prime spectrum.


If R is a commutative ring with identityPlanetmathPlanetmath and I is an ideal of R with V(I)=, then I=R.


Let R be a commutative ring with identity and I be an ideal of R with IR. Then, by this theorem (http://planetmath.org/EveryRingHasAMaximalIdeal), there exists a maximal idealMathworldPlanetmathPlanetmath M of R containing I. Since M is , then M is a proper prime idealMathworldPlanetmathPlanetmath of R. Thus, MV(I). The theorem follows. ∎

Title V(I)= implies I=R
Canonical name VIemptysetImpliesIR
Date of creation 2013-03-22 16:07:43
Last modified on 2013-03-22 16:07:43
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 10
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 14A15
Related topic ProofThatOperatornameSpecRIsQuasiCompact