Abstract: To check that every ideal of a commutative ring is finitely generated, it is enough to test the prime ideals. To check that every ideal is principal, again it is enough to test the prime ideals. We will show that neither statement is true if the word "prime" is changed to "maximal"; using freshman calculus we will construct a commutative ring in which every maximal ideal is principal but in which certain ideals are not even finitely generated, thus establishing that calculus has important practical applications. This talk is geared toward graduate students (though faculty are obviously welcome too) and will assume only a knowledge of calculus and some basic commutative ring theory.
