Modules with ascending chain condition on annihilators and Goldie modules

• Published in: arXiv.org 2016-01
• Main Authors: Jaime Castro Pérez, Mauricio Medina Bárcenas, José Ríos Montes
• Format: Article
• Language: English
• Subjects: Chains; Isomorphism; Modules
• ISSN: 2331-8422

Using the concepts of prime module, semiprime module and the concept of ascending chain condition (ACC) on annihilators for an \(R\)-module \(M\) . We prove that if \ \(M\) is semiprime \ and projective in \(\sigma \left[ M\right] \), such that \(M\) satisfies ACC on annihilators, then \(M\) has finitely many minimal prime submodules. Moreover if each submodule \(N\subseteq M\) contains a uniform submodule, we prove that there is a bijective correspondence between a complete set of representatives of isomorphism classes of indecomposable non \(M\)-singular injective modules in \(\sigma \left[ M\right] \) and the set of minimal primes in \(M\). If \(M\) is Goldie module then \(% \hat{M}\cong E_{1}^{k_{1}}\oplus E_{2}^{k_{2}}\oplus ...\oplus E_{n}^{k_{n}}\) where each \(E_{i}\) is a uniform \(M\)-injective module. As an application, new characterizations of left Goldie rings are obtained.