Amalgamated Products of Ore and Quadratic Extensions of Rings

TL;DR

This paper investigates the ideal structure of amalgamated products of Ore and quadratic extensions over rings, establishing conditions for principal ideals and applying results to the double affine Hecke algebra of GL_2(k).

Contribution

It introduces an analogue of the Hilbert Basis theorem for amalgamated products and characterizes when such products are principal ideal rings, with applications to Hecke algebras.

Findings

Proved an analogue of the Hilbert Basis theorem for amalgamated products.

Identified conditions for one-sided ideals to be principal or doubly-generated.

Showed that the double affine Hecke algebra H_{q,t} is a noetherian ring and an amalgamated product of quadratic extensions.

Abstract

We study the ideal theory of amalgamated products of Ore and quadratic extensions over a base ring R. We prove an analogue of the Hilbert Basis theorem for an amalgamated product Q of quadratic extensions and determine conditions for when the one-sided ideals of Q are principal or doubly-generated. We also determine conditions that make Q a principal ideal ring. Finally, we show that the double affine Hecke algebra $H_{q,t}$ associated to the general linear group GL_2(k) (here, k is a field with characteristic not 2) is an amalgamated product of quadratic extensions over a three-dimensional quantum torus and give an explicit isomorphism. In this case, it follows that $H_{q,t}$ is a noetherian ring.