Amitsur--Small Extensions and a Skew Amitsur--Small Theorem

TL;DR

The paper generalizes the Amitsur--Small theorem to skew settings by introducing Amitsur--Small extensions based on skew polynomial rings in several variables. It defines the AS-extension property, proves that it forces nontrivial intersection with the base ring for maximal left ideals, and identifies broad classes of rings (notably PIDs with many invariant maximal ideals) that yield AS-extensions. This framework leads to a skew Amitsur--Small theorem: if the center of $D[x_1, cdots,x_n;\sigma,\delta]$ contains a nonconstant polynomial, then every simple module over the skew polynomial ring is finite-dimensional over $D$. The paper also connects these skew results to well-known results about centers and maximal ideals, and provides concrete examples and corollaries, including finite-field cases with automorphisms of finite order.

Abstract

We introduce the notion of Amitsur--Small extensions to generalize a key lemma underlying the Amitsur--Small Theorem to the skew setting. Building on this framework, we establish a skew version of the Amitsur--Small Theorem.

Theorems & Definitions (30)

• Theorem 1.1: Amitsur--Small Theorem
• Lemma 1.2: Lemma C in AmitsurSmall
• Theorem 1.3: Skew Amitsur--Small Theorem
• Proposition 1.4
• Definition 2.1
• Example 2.1
• Example 2.2
• Proposition 2.1
• proof
• Definition 3.1