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Browsing Department of Mathematics by Author "Kumar, Rahul"

Browsing Department of Mathematics by Author "Kumar, Rahul"

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  • Almost ϕ-integrally closed rings 
    Kumar, Rahul (Taylor & Francis, 2023-09)
    Let 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 ...
  • Avoidance Principle and Intersection Property for a Class of Ring 
    Kumar, Rahul (Springer, 2020-04)
    Let R be a commutative ring with identity. If a ring R is contained in an arbitrary union of rings, then R is contained in one of them under various conditions. Similarly, if an arbitrary intersection of rings is contained ...
  • Chiral Metafilms and Surface Enhanced Raman Scattering for Enantiomeric Discrimination of Helicoid Nanoparticles 
    Kumar, Rahul (Wiley, 2023)
    Chiral nanophotonic platforms provide a means of creating near fields with both enhanced asymmetric properties and intensities. They can be exploited for optical measurements that allow enantiomeric discrimination at ...
  • Comment on “Two notes on imbedded prime divisors 
    Kumar, Rahul (ARXIV, 2020)
    The following result was proved in [5,Remark 2.2]. Theorem 0.1. If R T are Noetherian rings such that there does not exist any integrally dependent adjacent Noetherian rings between them, then for each ¯c/¯b 2 T/Z ...
  • Comparable overrings of a commutative ring 
    Kumar, Rahul (Springer, 2023-12)
    Let 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 ...
  • A Corrigendum to “Hereditary Properties Between a Ring and Its Maximal Subrings” 
    Kumar, Rahul (Springer, 2018-11)
    Let R be a commutative ring with identity. In A. Azarang, O. A. S. Karamzadeh, and A. Namazi, [Ukr. Math. J., 65, No. 7, 981–994 (2013) (Proposition 3.1)], it was proved that if R is an integral domain and S is a maximal ...
  • Maximal non valuation domains in an integral domain 
    Kumar, Rahul (Springer, 2020)
    Let R be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring R of an integral domain S is called a maximal non valuation domain ...
  • Maximal non valuative domains 
    Kumar, Rahul (ARXIV, 2020-09)
    The notion of maximal non valuative domain is introduced and characterized. An integral domain R is called a maximal non valuative domain if R is not a valuative domain but every proper overring of R is a valuative ...
  • Maximal non λ-subrings 
    Kumar, Rahul (Springer, 2020)
    Let R be a commutative ring with unity. The notion of maximal non -subrings is introduced and studied. A ring R is called a maximal non -subring of a ring T if R T is not a -extension, and for any ring S such that ...
  • Maximal Non ϕ -Chained Rings and Maximal Non Chained Rings 
    Kumar, Rahul (Springer, 2019-06)
    Let R be a commutative ring with unity. The notion of maximal non chained subrings of a ring and maximal non ϕ-chained subrings of a ring is introduced which generalizes the concept of maximal non valuation subrings of a ...
  • Maximal non-nonnil-principal ideal rings 
    Kumar, Rahul (World Scientific, 2025)
    Let 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 ...
  • Maximal non-Prüfer and maximal non--Prüfer rings 
    Kumar, Rahul (Taylor & Francis, 2021-10)
    Let R be a commutative ring with unity. Let H denotes the set of all rings R such that Nil(R) is a divided prime ideal. The notion of maximal non-Prüfer ring and maximal non-ϕ-Prüfer ring is introduced which generalize the ...
  • Maximal non-pseudovaluation subrings of an integral domain 
    Kumar, Rahul (Springer, 2024-06)
    The 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 ...
  • Maximal non-ϕ-pseudo-valuation rings 
    Kumar, Rahul (World Scientific, 2022)
    The notion of maximal non-ϕ-pseudo-valuation ring is introduced which generalizes the concept of maximal non-pseudo-valuation domain. The equivalence of maximal non-ϕ-PVR and maximal non-local ring is established under ...
  • A note on -domains and -domains 
    Kumar, Rahul (The Belgian Mathematical Society, 2020)
    Let R be an integral domain. Then R is said to be a λ-domain if the set of all overrings of R is linearly ordered by inclusion. If R1+R2 is an overring of R for each pair of overrings R1,R2 of R, then R is said to be a ...
  • A Note on FMS Modules and FCP Extensions 
    Kumar, Rahul (Springer, 2022-12)
    Let 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 ...
  • A note on imbedded prime divisors 
    Kumar, Rahul (Springer, 2020-06)
    In this note, we show that a part of Ratliff (Proc Am Math Soc 101(3):395–402, 1987, Remark 2.2) is not correct. Some conditions are given under which the same holds.
  • A note on maximal non-λ -rings 
    Kumar, Rahul (World Scientific, 2023)
    Let 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.
  • A note on maximal non-λ -rings 
    Kumar, Rahul (World Scientific, 2023)
    Let 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.
  • A note on pairs of rings with the same prime ideals 
    Kumar, Rahul (Hiroshima University, 2022-03)
    We study the ring extensions R⊆T having the same set of prime ideals provided Nil(R) is a divided prime ideal. Some conditions are given under which no such T exists properly containing R. Using idealization theory, the ...

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