The Structure of the Reduced Product of Preinjective Modules Over a Hereditary Right Pure Semisimple Ring

Abstract

ABSTRACT Let G be a division ring and F ⊂ G a fixed subdivision ring of G such that dim F G = ∞, dim G F = 2 and the F-G-bimodule F G G has the infinite dimension type d−∞( F G G ) = (…, 2, 2…,2, 2, 1, ∞) in the sense of Simson (19962000), see Section 2. In particular, the hereditary ring is a counter-example to the pure semisimplicity conjecture. Let be a complete set of representatives of pairwise non-isomorphic preinjective right R G -modules. We prove that the reduced product is a direct sum , where , , s 0 = s 1 = (card G)ℵ0 , and means the direct sum of s j copies of the right ideal P j of R G . We also show that is isomorphic to ∏ℵ0 P 1. Communicated by J. Gomaz Pardo.