Nonnil-FP-injective and nonnil-FP-projective dimensions and nonnil-semihereditary rings

In this paper, we introduce the concept of nonnil-FP-injective dimension for both modules and rings. We explore the characterization of strongly φ ϕ -rings that have a nonnil-FP-injective dimension of at most one. We demonstrate that, for a nonnil-coherent, strongly φ ϕ -ring R , the nonnil-FP-injective dimension of R corresponds to the supremum of the φ ϕ -projective dimensions of specific families of R -modules. We also define self-nonnil-injective rings as φ ϕ -rings that act as nonnil semi-injective modules over themselves and establish the equivalence between a strongly φ ϕ -ring R being φ ϕ -von Neumann regular and R being both nonnil-coherent and self-nonnil semi-injective. Furthermore, we extend the notion of semihereditary rings to φ ϕ -rings, coining the term ‘nonnil-semihereditary’ to describe rings where every finitely generated nonnil ideal is u- φ ϕ -projective. We provide several characterizations of nonnil-semihereditary rings through various conceptual lenses. Our study also includes an investigation of the transfer of the nonnil-semihereditary property in trivial ring extensions. Additionally, we define the nonnil-FP-projective dimension for modules and rings, showing that for any strongly φ ϕ -ring, a nonnil-FP-projective dimension of zero is indicative of the ring being nonnil-Noetherian. We also ascertain that, for a strongly φ ϕ -ring R , its nonnil-FP-projective dimension is the supremum of the NFP-projective dimensions across different families of R -modules. Lastly, we provide numerous examples to illustrate our results.