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In this paper, we first introduce and study the notions of strongly $\phi$-flat modules and strongly nonnil-injective modules. And then, we investigate the homology dimensions of modules and rings in terms of these two notions. Finally we give a new homological characterizations of $\phi$-Dedekind rings and $\phi$-\Prufer\ rings., Comment: Add Theorem 1.8. arXiv admin note: text overlap with arXiv:2107.12643

Let $D$ be an integral domain. Then $D$ is an almost valuation (AV-)domain if for $a, b\in D\setminus \{0\}$ there exists a natural number $n$ with $a^{n}\mid b^{n}$ or $b^{n}\mid a^{n}$. AV-domains are closely related to valuation domains, for example, $D$ is an AV-domain if and only if the integral closure $\bar{D}$ is a valuation domain and $D\subseteq \bar{D}$ is a root extension. In this note we explore various generalizations of DVRs (which we might call almost DVRs) such as Noetherian AV-domains, AV-domains with $\bar{D}$ a DVR, and quasilocal and local API-domains (i.e., for $\{a_{\alpha}\}_{\alpha\in \Lambda}\subseteq D$, there exists an $n$ with $(\{a_{\alpha}^{n}\}_{\alpha\in \Lambda})$ principal). The structure of complete local AV-domains and API-domains is determined.

In this paper we introduce two new generalizations of Krull domains: $\ast$-almost independent rings of Krull type ($\ast$-almost IRKTs) and $\ast$-almost generalized Krull domains ($\ast$-AGKDs), neither of which need be integrally closed. We characterize them using certain types of $\ast$-homogeneous ideals. To do this we introduce $\ast$-almost super-homogeneous ideals and $\ast$-almost super-SH domains. We prove that a domain $D$ is a $\ast$-almost IRKT if and only if $D$ is a $\ast$-almost super-SH domain and that a domain is a $\ast$-AGKD if and only if $D$ is a type 1 $\ast$-almost super-SH domain. Further, we study $\ast$-almost factorial general-SH domains ($\ast$-afg SH domains) and we prove that a domain $D$ is a $\ast$-afg-SH domain if and only if $D$ is a $\ast$-IRKT and an AGCD-domain.

Let $R$ be a ring, and $n$ a fixed nonnegative integer. An $R$-module $W$ is called $L_{n}$-injective if ${\rm Ext}_{R}^{1}(M,W)=0$ for any $R$-module $M$ with flat dimension at most $n$. In this paper, we prove first that ($\mathcal{F}_{n},\mathcal{L}_{n}$) is a complete hereditary cotorsion theory, where $\mathcal{F}_n$ (resp. $\mathcal{L}_n$) denotes the class of all $R$-modules with flat dimension at most $n$ (resp. $L_{n}$-injective $R$-modules). Then we introduce the $L_{n}$-injective dimension of a module and $L_n$-global dimension of a ring. Finally, over rings with weak global dimension $\leq n$, perfect rings, and $L_n$-hereditary rings, more properties and applications of $L_{n}$-injective modules, $L_{n}$-injective dimensions of modules and $\mathcal{L}_{n}$-global dimensions of rings are given., Comment: 19 pages