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Item A generalization of conducive domains(The Korean Mathematical Society, 2024-11) Kumar, RahulA domain R is called conducive if every conductor ideal (R:T) is nonzero for all overrings T of R other than the quotient field of R. Let H denote the set of all commutative rings R for which the set of all nilpotent elements forms a divided prime ideal. We extend the concept of conducive domains to the rings in the class H. Initially, we explore the basic properties of ϕ-conducive rings and rings closely related to them. Subsequently, we study these properties in the context of a specific pullback construction and a trivial ring extension.Item Does energy poverty matter for decent employment among women in india? An empirical insight from the dissimilarity index(2025-02) Kumar, Rahul; Bal, Debi PrasadThis study examines the impact of multidimensional energy poverty on women’s access to regular employment opportunities in India and its major states. The World Bank’s “human opportunity index” framework is applied to a sample of 31,611 employed women from the fifth round of the National Family and Health Survey. The findings indicate that energy poverty significantly affects women’s access to regular employment.Item A question about maximal non φ-chained subrings(Korea Science, 2023) Kumar, RahulLet 𝓗0 be the set of rings R such that Nil(R) = Z(R) is a divided prime ideal of R. The concept of maximal non φ-chained subrings is a generalization of maximal non valuation subrings from domains to rings in 𝓗0. This generalization was introduced in [20] where the authors proved that if R ∈ 𝓗0 is an integrally closed ring with finite Krull dimension, then R is a maximal non φ-chained subring of T(R) if and only if R is not local and |[R, T(R)]| = dim(R) + 3. This motivates us to investigate the other natural numbers n for which R is a maximal non φ-chained subring of some overring S. The existence of such an overring S of R is shown for 3 ≤ n ≤ 6, and no such overring exists for n = 7.Item On minimal ring extensions(2020-05) Kumar, RahulLet R be a commutative ring with identity. The ring R×R can be viewed as an extension of R via the diagonal map Δ:R↪R×R, given by Δ(r)=(r,r) for all r∈R. It is shown that, for any a,b∈R, the extension Δ(R)[(a,b)]⊂R×R is a minimal ring extension if and only if the ideal is a maximal ideal of R. A complete classification of maximal subrings of R(+)R is also given. The minimal ring extension of a von Neumann regular ring R is either a von Neumann regular ring or the idealization R(+)R/m where m∈Max(R). If R⊂T is a minimal ring extension and T is an integral domain, then (R:T)=0 if and only if R is a field and T is a minimal field extension of R, or RJ is a valuation ring of altitude one and TJ is its quotient field.Item A note on maximal non-λ -rings(World Scientific, 2023) Kumar, RahulLet ℋ0 denote the set of all rings R such that Nil(R) is a divided prime ideal with Nil(R)=Z(R). We study the concept of maximal non-λ-rings in class ℋ0 and generalize the results of maximal non-λ-domains.Item Comparable overrings of a commutative ring(Springer, 2023-12) Kumar, RahulLet H be the set of all commutative rings R such that Nil(R) is a divided prime ideal of R and let φ : T (R) → RNil(R) be a ring homomorphism defined as φ(x) = x for all x ∈ T (R). An overring Ro of an integral domain R is said to be comparable if Ro = R, Ro = qf(R), and each overring of R is comparable to Ro under inclusion. We study comparable overrings of a ring in class H.Item Almost ϕ-integrally closed rings(Taylor & Francis, 2023-09) Kumar, RahulLet R be a commutative ring with unity. The notion of almost 𝜙-integrally closed ring is introduced which generalizes the concept of almost integrally closed domain. Let ℋ be the set of all rings such that Nil(𝑅) is a divided prime ideal of R and 𝜙:𝑇(𝑅)→𝑅Nil(𝑅) is a ring homomorphism defined as 𝜙(𝑥)=𝑥 for all 𝑥∈𝑇(𝑅). A ring 𝑅∈ℋ is said to be an almost 𝜙-integrally closed ring if 𝜙(𝑅) is integrally closed in 𝜙(𝑅)𝜙(𝔭) for each nonnil prime ideal 𝔭 of R. Using the idealization theory of Nagata, examples are also given to strengthen the concept.Item Maximal non-nonnil-principal ideal rings(World Scientific, 2025) Kumar, RahulLet H be the set of all commutative rings with unity whose nilradical is a divided prime ideal. The concept of maximal non-nonnil-PIR is introduced to generalize the concept of maximal non-PID. A ring extension R⊂T in H is a called a maximal non-nonnil-principal ideal ring if R is not a nonnil-principal ideal ring but each subring of T properly containing R is a nonnil-principal ideal ring. It is shown that R+XT[X] (respectively, R+XT[[X]]) is a maximal non-nonnil-PIR subring of T[X] (respectively, T[[X]]) if and only if R+XT[X] (respectively, R+XT[[X]]) is a maximal non-PID subring of T[X] (respectively, T[[X]]).Item Maximal non-pseudovaluation subrings of an integral domain(Springer, 2024-06) Kumar, RahulThe notion of maximal non-pseudovaluation subring of an integral domain is introduced and studied. Let R ⊂ S be an extension of domains. Then R is called a maximal non-pseudovaluation subring of S if R is not a pseudovaluation subring of S, and for any ring T such that R ⊂ T ⊂ S, T is a pseudovaluation subring of S. We show that if S is not local, then there no such T exists between R and S. We also characterize maximal non-pseudovaluation subrings of a local integral domain.Item Single crystalline Ge thin film growth on c-plane sapphire substrates by molecular beam epitaxy (MBE)(RSC, 2022-04) Kumar, RahulSingle crystalline Ge has been grown on c-plane sapphire substrates by molecular beam epitaxy. Direct growth of Ge on sapphire results in three-dimensional (3D) Ge islands, two growth directions, more than one primary domain, and twinned crystals. The introduction of a thin AlAs nucleation layer significantly improved the surface and material quality, which is evident from a smoother surface, single epitaxial orientation, sharper rocking curve, and a single domain. The AlAs nucleation layer thickness was also investigated, and a 10 nm AlAs layer resulted in the lowest surface roughness of 3.9 nm. We have been able to achieve a single primary domain and reduced twinning relative to previous works. A high-quality Ge buffer on sapphire has the potential as an effective platform for the subsequent growth of GeSn and SiGeSn for microwave photonics.