A Note on Weakly Semiprime Ideals and Their Relationship to Prime Radical in Noncommutative Rings

In this paper, we introduce the concept of weakly semiprime ideals and weakly -systems in noncommutative rings. We establish the equivalence between an ideal being a weakly semiprime ideal and being a weakly -system. We provide alternative definitions and explore the properties of weakly semiprime ideals. Additionally, we investigate scenarios where all ideals in a given ring are weakly semiprime and demonstrate that in Noetherian rings, where every ideal is weakly semiprime, the prime radical and the Jacobson radical coincide.