S-versions and S-generalizations of nil-clean ideals and Nil*-Noetherian rings

We propose the idea of [Formula: see text]-nil-clean ideals in this work. We prove that Jacobson’s Lemma holds for [Formula: see text]-nil-clean elements. Furthermore, we define the [Formula: see text]-version of a Boolean ring and prove that every ideal of an [Formula: see text]-Boolean ring is an [Formula: see text]-nil-clean ideal. In addition, we show that a class of von Neumann regular local rings with maximal ideal [Formula: see text] disjoint from [Formula: see text] belongs to the class of [Formula: see text]-clean rings. Subsequently, we define the concept of a Nil[Formula: see text]-[Formula: see text]-Noetherian ring, which serves as a proper generalization of both Nil[Formula: see text]-Noetherian and [Formula: see text]-Noetherian rings. Among our main results, we demonstrate the existence of [Formula: see text]-primary decomposition in Nil[Formula: see text]-[Formula: see text]-Noetherian rings, thereby generalizing the classical Lasker–Noether theorem.