On Strongly Nonnil-Coherent Rings and Strongly Nonnil-Noetherian Rings

Abstract. In this paper, all rings considered are assumed to be commutative with non-zero identity and prime nilradical. We define a subclass of nonnil-cohrent rings introduced by the authors of [1] which we will call strongly nonnil-coherent. Following the work of the authors of [9], we introduce the homological methods of the strongly nonnil-coherent rings as in the classical case. Recall from [4] that a φ-ring R is said to be nonnil-Noetherian if every nonnil ideal of R is finitely generated, which is equivalent to saying that a φ-ring R is called nonnil-Noetherian if R/Nil(R) is a Noetherian domain [4, Theorem 1.2]. Notice that a nonnil-Noetherian ring is not necessarily nonnil-coherent as shown in [8, Example 4.11]. However, in the least section of this paper, we introduce and study a new sub-class of nonnil-Noetherian rings which is contained in the class of strongly nonnil-coherent rings.