# 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?