# Math Help - maximal ideal in commutative algebra

1. ## maximal ideal in commutative algebra

I would like to show that every commutative algebra contains proper maximal ideals

My plan is to create en increasing sequence of proper ideals in the commutative algebra $A$ of the sort $I_1\subseteq I_2\subseteq\dots$ such that $I=\bigcup^{\infty}_{n=1}I_{n}$
then to show that every other porper ideal $J$ of $A$ will be contained in the union of some finite collection of these proper ideals $I_{n}$ then $I$ will be a maximal ideal.

My problem is that I didn't really make any use of the commutative structure of the algebra, any ideas?

2. Originally Posted by Mauritzvdworm
I would like to show that every commutative algebra contains proper maximal ideals

My plan is to create en increasing sequence of proper ideals in the commutative algebra $A$ of the sort $I_1\subseteq I_2\subseteq\dots$ such that $I=\bigcup^{\infty}_{n=1}I_{n}$
then to show that every other porper ideal $J$ of $A$ will be contained in the union of some finite collection of these proper ideals $I_{n}$ then $I$ will be a maximal ideal.

My problem is that I didn't really make any use of the commutative structure of the algebra, any ideas?
See link1 and link2.

3. Notice also that when you are taking the union of an increasing chain of proper ideals, you need to ensure that the union is still a proper ideal. If the algebra is unital then this is not a problem, since none of the ideal contains the identity element and so neither does their union. But in the case of a nonunital algebra you need to use the concept of a modular ideal.

It does not matter that you didn't make any use of the commutative structure of the algebra, because the result is equally valid in the case of a noncommutative algebra (though in that case you need to specify whether you are using left, right, or two-sided, ideals).