Maximal ideals

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  • Definition of a maximal ideal

An ideal [math]M[/math] of a commutative ring [math]R[/math] is a maximal ideal of [math]R[/math] if

(1) [math]M \neq R[/math];

(2) if [math]I[/math] is an ideal of [math]R[/math] with [math]M \subseteq I \subseteq R[/math] then either [math]I = M[/math] or [math]I = R[/math].

  • Theorem

Let [math]R[/math] be a commmutative ring and let [math]M[/math] be an ideal of [math]R[/math]. Then [math]M[/math] is a maximal ideal of [math]R[/math] if and only if the factor ring [math]R/M[/math] is a field.

Back to Chapter 1: Basics of Ring Theory