Nonnil-FP-injective and nonnil-FP-projective dimensions and nonnil-semihereditary rings
Abstract
In this paper, we introduce the concept of nonnil-FP-injective dimension for both modules and rings. We explore the characterization of strongly ϕ -rings that have a nonnil-FP-injective dimension of at most one. We demonstrate that, for a nonnil-coherent, strongly ϕ -ring R, the nonnil-FP-injective dimension of R corresponds to the supremum of the ϕ -projective dimensions of specific families of R-modules. We also define self-nonnil-injective rings as ϕ -rings that act as nonnil semi-injective modules over themselves and establish the equivalence between a strongly ϕ -ring R being 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 -rings, coining the term `nonnil-semihereditary' to describe rings where every finitely generated nonnil ideal is u-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 -ring, a nonnil-FP-projective dimension of zero is indicative of the ring being nonnil-Noetherian. We also ascertain that, for a strongly 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.
- Publication:
-
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas
- Pub Date:
- May 2024
- DOI:
- 10.1007/s13398-024-01604-0
- Bibcode:
- 2024RRACE.118..104H
- Keywords:
-
- Nonnil-coherent ring;
- ϕ -submodule;
- Nonnil-FP-injective dimension;
- Nonnil-FP-projective dimension;
- Nonnil-semihereditary ring;
- ϕ -von Neumann regular ring