Uniform Modules Over Goldie Prime Serial Rings

Abstract

<p>We investigate the uniseriality of uniform modules. Let R be any ring and fix a decomposition 1 = ℯ₁ + ℯ₂ +…+ℯn into orthogonal idempotents. Let Vʀ be uniform and injective; we prove that there exists ℯ = ℯᵢ such that VR ≅ homA (Rℯ, Vℯ) where A = ℯRℯ. Moreover, Vℯ is a uniform injective A-module. If R is Goldie prime serial, we prove that V is uniserial if and only if Vℯ is uniserial as an A-module.</p> <p>If R is Goldie prime serial, we know that such an A is a valuation on a division ring D. We prove that any uniform injective, EA , is of the form E = E (D/I) for some I ≤ A. If D/I is injective, then E is uniserial. We give several necessary and sufficient conditions for D/I to be injective.</p> <p>In this study of uniform injectives over Goldie prime serial rings we define a notion of generalized associated primes. This leads to a semiprime Goldie ideal, S, which can be associated to any uniform injective. We prove that for certain uniform electives, C(S) (the set of elements regular modulo S) is the largest Ore set operating regularly on the module.</p>