A counterexample to the pure semisimplicity conjecture

A ring is right pure semisimple if every right module is a direct sum of finitely generated modules, and the pure semisimplicity conjecture asserts that every such ring is of finite representation type. In this article, a division ring extension H⊆ T is constructed such that the bimodule HTT has right dimension sequence (1,2,2,2,) and T is infinite-dimensional as a left H-space. The division ring T is the union of the fraction division rings of the nearest-neighbour quantum Laurent algebras with parameter $1/2$, and H is the image of T under a compatible family of conjugations. The extension is controlled by right bases of every length and by two-sided flags of adjacent division subrings, which identify all iterated right duals up to explicit twists. It follows that the triangular ring (smallmatrixH&T\\0&Tsmallmatrix) is right pure semisimple but neither left pure semisimple nor of finite representation type, so that the pure semisimplicity conjecture fails.