On a problem of Amitsur and Small

TL;DR

This paper constructs a specific example of a division ring and a maximal left ideal in a polynomial ring, resolving a longstanding problem posed by Amitsur and Small in 1978.

Contribution

It provides the first known example demonstrating that the intersection of a maximal left ideal in a polynomial ring over a division ring can fail to be maximal in the polynomial subring.

Findings

Constructed a division ring D and a maximal left ideal M in D[x,y]

Showed M ∩ D[x] is not maximal in D[x]

Resolved a problem posed by Amitsur and Small in 1978

Abstract

We construct an example of a division ring $D$ and a maximal left ideal $M$ in the polynomial ring $D[x,y]$ in two central variables over $D$, such that the intersection $M \cap D[x]$ is not a maximal left ideal in $D[x]$. This resolves a ring-theoretic problem of Amitsur and Small raised in 1978.