The aim of this research paper was to explore the various characteristics of the doubly warped product manifold, focusing particularly on aspects such as the Hessian, Riemannian curvature, Ricci curvature, and concircular curvature tensor components. By examining the necessary conditions that would classify the manifold as Riemann-flat, Ricci-flat, and concircularly-flat, the study aimed to expand our understanding of these concepts. To achieve this, the research incorporated the application of these findings to a generalized Robertson-Walker doubly warped product manifold scenario. This approach allowed us to identify and analyze the specific circumstances under which the manifold displayed concircular flatness.

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. In this paper we investigate the class of QTAG-modules whose every separable module is a direct sum of uniserial modules; such modules are called ω-totally ω-projective. We discuss an interesting characterization of this class and we show that the class of $(\omega +n)$-totally $(\omega +n)$-projective modules contains the class of ω-totally ω-projective modules.

A module \(M\) over an associative ring with unity is a \(QTAG\) -module if every finitely generated submodule of any homomorphic image of \(M\) is a direct sum of uniserial modules. There are many fascinating concepts related to these modules. Here we introduce the notion of \(n\) -layered \(QTAG\) -modules and discuss some interesting properties of these modules. We show that a \(QTAG\) -module \(M\) is \(n\) -layered if and only if \(M/N\) is an \(n\) -layered module, whenever \(N\) is a finitely generated submodule of \(M\) and \(n\ge 1\) is an integer.

This study presents new fuzzy adaptations of Ostrowski’s integral inequalities through a novel class of convex fuzzy-valued mappings defined over a harmonic convex set, avoiding the use of the Sugeno integral. These innovative inequalities generalize the recently developed interval forms of real-valued Ostrowski inequalities. Their formulations incorporate integrability concepts for fuzzy-valued mappings (FVMs), applying the Kaleva integral and a Kulisch–Miranker fuzzy order relation. The fuzzy order relation is constructed via a level-wise approach based on the Kulisch–Miranker order within the fuzzy number space. Additionally, numerical examples illustrate the effectiveness and significance of the proposed theoretical model. Various applications are explored using different means, and some complex cases are derived.

In this paper, we generalize a suitable transformation from an element-based to a submodule-based interpretation of the traditional idea of transitivity in QTAG modules. We examine QTAG modules that are transitive in the sense that the module has an automorphism that sends one isotype submodule $ K $ onto any other isotype submodule $ K' $, unless this is impossible because either the submodules or the quotient modules are not isomorphic. Additionally, the classes of strongly transitive and strongly $ U $-transitive QTAG modules are defined using a slight adaptations of this. This work investigates the latter class in depth, demonstrating that every $ \alpha $- module is strongly transitive with regard to countably generated isotype submodules.