# maximal ideal in commutative algebra

• Mar 6th 2010, 02:59 AM
Mauritzvdworm
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 $\displaystyle A$ of the sort $\displaystyle I_1\subseteq I_2\subseteq\dots$ such that $\displaystyle I=\bigcup^{\infty}_{n=1}I_{n}$
then to show that every other porper ideal $\displaystyle J$ of $\displaystyle A$ will be contained in the union of some finite collection of these proper ideals $\displaystyle I_{n}$ then $\displaystyle I$ 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?
• Mar 6th 2010, 10:19 PM
aliceinwonderland
Quote:

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 $\displaystyle A$ of the sort $\displaystyle I_1\subseteq I_2\subseteq\dots$ such that $\displaystyle I=\bigcup^{\infty}_{n=1}I_{n}$
then to show that every other porper ideal $\displaystyle J$ of $\displaystyle A$ will be contained in the union of some finite collection of these proper ideals $\displaystyle I_{n}$ then $\displaystyle I$ 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?