From: Fred W. Helenius Subject: Re: What is the maximal ideal of even rings? Date: Tue, 24 Oct 2000 23:28:44 -0400 Newsgroups: sci.math Summary: [missing] "Chip Eastham" wrote: >Virgil explicitly demonstrates toward the beginning of this thread >that the particular "rng" (ring without unity, not random number >generator!) has a maximal ideal. However, IIRC, the general proof >that a ring with unity must have a maximal ideal depends on Zorn's >lemma and the fact that we can maximize ideals with respect to the >exclusion of the identity element 1. >Does anyone know whether this is generally true of rngs, i.e. a >maximal ideal must exist? [I got started on a similar train of >thought about maximal subgroups in infinite simple groups >from reading another thread in this NG.] I'd guess that >it is not true, but I don't yet have a "counterexample". There's a useful trick for generating counterexamples among rings without unity: find an abelian group with the corresponding property and make it into a ring by using the trivial multiplication in which all products are zero; the ideals of the ring are just the subgroups of the group. In this case, we want an abelian group with no maximal proper subgroup, and I remember an example: Let R consist of the dyadic rationals (i.e., those with denominator a power of 2) in the interval [0,1), with addition defined by addition modulo 1, and with trivial multiplication. Note that any element of the form m/2^k with m odd generates 1/2^k, and thus all of the elements with denominator 2^k. Given any ideal, the denominators of its elements (in lowest terms) are either bounded or not; if they are unbounded, the ideal is the entire ring, otherwise it is the ideal I_k generated by 1/2^k for some k. The larger k is, the larger the ideal; so there is no maximal proper ideal. -- Fred W. Helenius