Simplifying Smoktunowicz’s Extraordinary Example

At the turn of the 21st century Agata Smoktunowicz constructed the first example of a nil algebra over a countable field such that the polynomial ring over the algebra is not nil. This answered an old question of Amitsur. We present a simplification of the example. Introduction This monograph began as the author’s earnest attempt to understand the beautiful example constructed in [1], and was originally meant only to serve as a set of personal notes. However, the author believes the simplifications offered will draw more people to understand and appreciate Smoktunowicz’s example. Further, as of the writing of this paper the example has been used in more than twenty other research papers, and continues to be generalized and expanded. It is hoped that this note will aid in this process and simplify future constructions. Almost all of the original research comes from Smoktunowicz’s work in [1], and many of the lemmas, theorems, and corollaries of this note are direct generalizations of results from that paper. The present paper reworks and streamlines the bounds and numerical computations of that paper. Further, following the methods of [2], we answer the implicit question left open at the end of the paper, showing that the main construction technique needs only one polynomial variable (and not two) to reach the desired example. Rather than construct each of the components of the example separately and then put them together at the end, we take a motivational approach where we give the underlying reason for each piece before it is constructed. It is hoped that this will aid the reader in keeping track of the important information as it is needed. As the paper is in essence a single example, any quantities defined outside of lemmas, theorems, or corollaries will continue to be defined throughout the paper and retain their meanings when invoked in proofs or in the statement of propositions. 1. Nilpotence over polynomial rings Let K be a countable field. Our goal is to construct a nil K-algebra R such that the polynomial ring R[X] is not nil. The first half of our goal is simple. We let A = K〈x1, x2, x3〉 be the (unital) free algebra in the three noncommuting variables x1, x2, and x3. We also let A ′ denote the (nonunital) Ksubalgebra of A consisting of the elements with zero constant term. We fix an indexing {f1, f2, f3, . . .} of A′ by the positive integers. This is the one and only place where we use the countability of K. For each i ≥ 1 we let ei be a positive integer and we let I be an ideal of A containing fi i for every i ≥ 1. We set R = A′/I and see that R is nil. As arbitrary ideals are difficult to work with, we take the simplifying position that I is a homogeneous ideal. Thus, we assume: (1) The ideal I is generated by the homogeneous components of {fi i : i ≥ 1}. One might ask: Why use three noncommuting variables to form A? It turns out that in fact two variables suffice (which will be shown at the end of this section), but with three the proof is significantly 2010 Mathematics Subject Classification. Primary 16N40, Secondary 16N20, 16S36.