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

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 (1996 2000), 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.

Key Words:

• Preinjective modules
• Product conjecture
• Pure semisimple conjecture

Mathematics Subject Classification:

• 16K40
• 16G10

ACKNOWLEDGMENT

We thank Professor Daniel Simson for his encouragement. We also thank the referee for suggestions that influenced the final form of the article. Supported by MREEF.