On Soft Near‐Prime Int‐Ideals and Soft 1‐Absorbing Prime Int‐Ideals With Applications

In this study, we aimed to introduce two different generalizations of the soft prime int‐ideal and clarify the relationships between the soft prime int‐ideal and the substructures of a ring. First, we explored new algebraic features of the soft prime int‐ideal. We defined and exemplified essential concepts such as soft nilpotent elements, soft idempotent elements, nonexplicit soft int‐ideals, one‐to‐one soft sets, and soft zero divisors of a soft int‐ideal. Then, we investigated the behavior of soft nilpotent and soft idempotent elements in relation to the soft prime int‐ideal. We then defined a soft near‐prime int‐ideal, which is a generalization of the soft prime int‐ideal through soft radicals, and examined its properties. Furthermore, we analyzed the relationships between the soft prime int‐ideal and the soft near‐prime int‐ideal, showing that every soft prime int‐ideal is a soft near‐prime int‐ideal, but the converse does not hold. In addition, we proposed the concept of a soft 1‐absorbing prime int‐ideal as another generalization of the soft prime int‐ideal and studied its basic properties. We proved that the radicals of soft near‐prime int‐ideals and soft 1‐absorbing prime int‐ideals are soft prime int‐ideals. Lastly, we examined soft homomorphic images and preimages of these new structures.