Modules with ascending chain condition on annihilators and Goldie modules

الأصل · EN

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 σ[M], such that M satisfies ACC on annihilators, then M has finitely many minimal prime submodules. Moreover if each submodule N 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 σ[M] and the set of minimal primes in M. If M is Goldie module then % M E₁ᵏ₁⊕ E₂ᵏ₂⊕...⊕ Eₙᵏₙ where each Eᵢ is a uniform M-injective module. As an application, new characterizations of left Goldie rings are obtained.