  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.
