In this paper, we introduce ∗-almost independent rings of Krull type (∗-almost IRKTs) and ∗-almost generalized Krull domains (∗-almost GKDs) in the general theory of almost factoriality, neither of which need be integrally closed. This fills a gap left in [D. D. Anderson and M. Zafrullah, On∗-Semi-Homogeneous Integral Domains, Advances in Commutative Algebra (Springer, Singapore, 2019)]. We characterize them by ∗-almost super-SH domains, where a domain $D$ is called a ∗-almost super-SH domain if every nonzero proper principal ideal of $D$ is a ∗-product of ∗-almost super-homogeneous ideals. We prove that (1) a domain $D$ is a ∗-almost IRKT if and only if $D$ is a ∗-almost super-SH domain, (2) a domain is a ∗-almost GKD if and only if $D$ is a type 1 ∗-almost super-SH domain and (3) a domain $D$ is a ∗-almost IRKT and an AGCD-domain if and only if $D$ is a ∗-afg-SH domain. Further, we characterize them by their integral closures. For example, we prove that a domain $D$ is an almost IRKT if and only if $D\subseteq \overline{D}$ is a root extension with $D$$t$-linked under $\overline{D}$ and $\overline{D}$ is an IRKT. Examples are given to illustrate the new concepts.

在本文中，我们在几乎阶乘的一般理论中引入了 ∗-几乎独立的 Krull 型环 (∗-almost IRKTs) 和 ∗-almost 广义 Krull 域 (∗-almost GKDs)，两者都不需要整体闭合。这填补了 [DD Anderson 和 M. Zafrullah, On ∗ -Semi-Homogeneous Integral Domains , Advances in Commutative Algebra (Springer, Singapore, 2019)] 中留下的空白。我们用 ∗-almost super-SH 域来表征它们，其中一个域$D$如果每个非零真主理想$D$是*-几乎超齐次理想的*-乘积。我们证明 (1) 一个域$D$是一个 ∗-几乎 IRKT 当且仅当$D$是一个 ∗-almost 超 SH 域，(2) 一个域是一个 ∗-almost GKD 当且仅当$D$是一个类型 1 ∗-几乎是超 SH 域和 (3) 一个域$D$是一个 ∗-几乎 IRKT 和一个 AGCD 域当且仅当$D$是一个 *-afg-SH 域。此外，我们通过它们的整体闭包来表征它们。例如，我们证明一个域$D$是一个几乎 IRKT 当且仅当$D\subseteq \overline{D}$是一个根扩展$D$$吨$-链接下$\overline{D}$和$\overline{D}$是一个 IRKT。举例说明新概念。