See link1 and link2.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 of the sort such that

then to show that every other porper ideal of will be contained in the union of some finite collection of these proper ideals then 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?