maximal ideal


Let R be a ring with identity. A proper left (right, two-sided) ideal 𝔪R is said to be maximal if 𝔪 is not a proper subsetMathworldPlanetmathPlanetmath of any other proper left (right, two-sided) ideal of R.

One can prove:

  • A left idealMathworldPlanetmathPlanetmath 𝔪 is maximal if and only if R/𝔪 is a simple left R-module.

  • A right ideal 𝔪 is maximal if and only if R/𝔪 is a simple right R-module.

  • A two-sided ideal 𝔪 is maximal if and only if R/𝔪 is a simple ringMathworldPlanetmath.

All maximal idealsMathworldPlanetmath are prime idealsMathworldPlanetmath. If R is commutativePlanetmathPlanetmathPlanetmath, an ideal 𝔪R is maximal if and only if the quotient ringMathworldPlanetmath R/𝔪 is a field.

Title maximal ideal
Canonical name MaximalIdeal
Date of creation 2013-03-22 11:50:57
Last modified on 2013-03-22 11:50:57
Owner djao (24)
Last modified by djao (24)
Numerical id 8
Author djao (24)
Entry type Definition
Classification msc 13A15
Classification msc 16D25
Classification msc 81R50
Classification msc 46M20
Classification msc 18B40
Classification msc 22A22
Classification msc 46L05
Related topic ProperIdeal
Related topic Module
Related topic Comaximal
Related topic PrimeIdeal
Related topic EveryRingHasAMaximalIdeal