A Simple Characterization of Commutative Rings without Maximal Ideals

Abstract

In a coLrse in abstract algebra in which the instructor presents a proof that each ideal in a ring with identity is contained in a maximal ideal, it is customary to give an example of a ring without maximal ideals. The usual example is a zero-ring whose additive group has no maximal subgroups (e.g., the additive group of (dyadic) rational numbers; actually any divisible group will do; see [1, p. 67]). This may leave the impression that all such rings are artificial or at least that they abound with divisors of 0. Below, I give a simple characterization of commutative rings withoLt maximal ideals and a class of examples of such rings, including some without proper divisors of 0. To back up the contention that this can be presented in such a course in abstract algebra, I outline proofs of some known theorems including a few properties of radical rings in the sense of Jacobson. The Hausdorff maximal principle states that every partially ordered set contains a maximal chain (i.e., a maximal linearly ordered subset). It is equivalent to the axiom of choice [4, Chapter XI]. Since the union of a maximal chain of proper ideals in a ring with identity is a maximal ideal, and since the union of a maximal chain of linearly independent stubsets of a vector space is a maximal linearly independent set, we have: