This research presents an in-depth study of a new concept, namely annihilator closed-injective modules (denoted by ac-injective modules), and clarifies its relationship with other modules. The main contributions of the research are as follows: First, utilizing specific types of modules that establish an equivalence between essential and ann-essential submodules. Second, formally introducing and analyzing the concept of ac-injective modules. Third, investigating the hereditary nature of this concept: demonstrating, through specific counterexamples, that ac-injectivity is not a hereditary property in general, and proving the transfer of the self-ac-injective property to specific types of submodules, such as fully invariant ann-closed submodules. Fourth, providing an in-depth study of the self-ac-injective property under direct sums, including counterexamples for general cases and identifying sufficient conditions for the property to transfer to the sum.