criterion for maximal ideal

Theorem.  In a commutative ring R with non-zero unity, an ideal π”ͺ is maximal if and only if

βˆ€a∈Rβˆ–π”ͺβ’βˆƒr∈R such that⁒  1+a⁒r∈π”ͺ. (1)

Proof.  1∘.  Let first π”ͺ be a maximal idealMathworldPlanetmath of R and  a∈Rβˆ–π”ͺ.  Because  π”ͺ+(a)=R,  there exist some elements  m∈π”ͺ  and  -r∈R  such that  m-a⁒r=1.  Consequently,  1+a⁒r=m∈π”ͺ.
2∘.  Assume secondly that the ideal π”ͺ satisfies the condition (1).  Now there must be a maximal ideal π”ͺβ€² of R such that


Let us make the antithesis that π”ͺβ€²βˆ–π”ͺ is non-empty.  Choose an element


By our assumption, we can choose another element r of R such that

s= 1+a⁒r∈π”ͺβŠ‚π”ͺβ€².

Then we have


which is impossible since with 1 the ideal π”ͺβ€² would contain the whole R.  Thus the antithesis is wrong and  π”ͺ=π”ͺ′  is maximal.

Title criterion for maximal ideal
Canonical name CriterionForMaximalIdeal
Date of creation 2013-03-22 19:10:40
Last modified on 2013-03-22 19:10:40
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Theorem
Classification msc 16D25
Classification msc 13A15
Related topic MaximalIdealIsPrime