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

Abstract

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.