Modules with ascending chain condition on annihilators and Goldie modules

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 $$ where each $E_{i}$ is a uniform $M$-injective module. As an application, new characterizations of left Goldie rings are obtained.