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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 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 A Note on FMS Modules and FCP Extensions(Springer, 2022-12) Kumar, RahulLet R be a commutative ring with unity and S be a (unital) subring of R such that R is integral over S and S⊆R has FCP. Let M be an R-module. For any submodule N of M, it is shown that R(+)N⊆R(+)M has FCP if and only if S(+)N⊆S(+)M has FCP. We also discuss FMS modules.Item Three open questions on residually small rings(Rocky Mountain Mathematics Consortium, 2020-02) Kumar, RahulRecently in 2018, four open questions were raised by Oman and Salminen (2018). We answer three of them in this article.