Rings Without Maximal Ideals

Abstract

It is traditional in an abstract algebra class to prove, using Zorn's lemma, that a ring with unit must have maximal ideals. Without a unit element this is not true, and here we present some commutative counterexamples. First we consider rings with trivial multiplication, i.e., those for which any product is zero. Then an ideal is just an additive subgroup, and we are seeking abelian groups without maximal subgroups. Such groups are easily characterized using the notion of divisibility. If G is an abelian group, written additively, and m is a positive integer, then denote by mG the set {mglg c G}. Then G is said to be divisible if mG = G for every positive integer m. It is easy to verify that the additive group Q of rational numbers is divisible and that every direct sum of divisible groups is divisible.