Maximal ideals

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 29 December 2011

Maximal ideals

Let R be a commutative ring. A maximal ideal is an ideal M such that R/M is a field.

For each a let ev a : [ x ] f ( x ) f ( a ) be the evaluation homomorphism.

The map { maximal ideals of [ x ] } a ker ( ev a ) = ( x - a ) is a bijection.

For each a = a 1 ... a n n let x 1 ... x n n f x 1 ... x n f a 1 ... a n be the evaluation homomorphism.

(Weak Nullstellensatz.) The map n { maximal ideals of x 1 ... x n } a ker ( ev a ) = x 1 - a 1 ... x n - a n is a bijection.

Proof.

A variety is a set V n such that V = a n f 1 ( a ) = 0 , ... , f r ( a ) = 0 , for some finite set of polynomials f 1 , ... , f r x 1 ... x n .

Let f 1 , ... , f r x 1 ... x n and let V = a n f 1 ( a ) = 0 , ... , f r ( a ) = 0 . The map V { maximal ideals of x 1 ... x n / f 1 ... f r } a ker ( ev a ) + f 1 ... f r is a bijection.

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