Department of Mathematics

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Almost ϕ-integrally closed rings
(Taylor & Francis, 2023-09) Kumar, Rahul
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 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.
Avoidance Principle and Intersection Property for a Class of Ring
(Springer, 2020-04) Kumar, Rahul
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 in R, then R contains one of them under various conditions.
Chiral Metafilms and Surface Enhanced Raman Scattering for Enantiomeric Discrimination of Helicoid Nanoparticles
(Wiley, 2023) Kumar, Rahul
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 detection levels > 6 orders of magnitude than is achieved with conventional chirally sensitive spectroscopic methods based on circularly polarized light. Here it is shown that surface enhanced Raman spectroscopy (SERS) is such a local probe of the near field environment. It is used to achieve enantiomeric discrimination of chiral helicoid nanoparticles deposited on left- and righthanded enantiomorphs of a chiral metafilm using an achiral molecule as a probe. “Hotter” electromagnetic (EM) hotspots are created for matched combinations of helicoid and metafilms (left-left and right-right), while mismatched combinations leads to significantly “cooler” electromagnetic hotspots. This large enantiomeric dependency on hotspot intensity is readily detected using SERS with the aid of an achiral Raman reporter molecule. In effect SERS is used to distinguish between the different EM environments of the plasmonic diastereomers produced by mixing chiral nanoparticles and metafilms. The work demonstrates that by combining chiral nanophotonic platforms with established SERS strategies new avenues in ultrasensitive chiral detection can be opened.
Comment on “Two notes on imbedded prime divisors
(ARXIV, 2020) Kumar, Rahul
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 (where Z = Rad(T) = Rad(R) and ¯b, ¯c regular in R/Z), we have either ¯c/¯b 2 R/Z or ¯ b/¯c 2 R/Z, and so (R/Z)[¯c/¯b] is a localization of R/Z.
Comparable overrings of a commutative ring
(Springer, 2023-12) Kumar, Rahul
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 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.
A Corrigendum to “Hereditary Properties Between a Ring and Its Maximal Subrings”
(Springer, 2018-11) Kumar, Rahul
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 subring of R integrally closed in R, then dim(S) = 1 implies that dim(R) = 1 if and only if (S : R) = 0. An example is given, which shows that the above-mentioned proposition is not true.
A generalization of conducive domains
(The Korean Mathematical Society, 2024-11) Kumar, Rahul
A 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.
Maximal non valuation domains in an integral domain
(Springer, 2020) Kumar, Rahul
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 in S if R is not a valuation subring of S, and for any ring T such that R T S, T is a valuation subring of S. For a local domain S, the equivalence of an integrally closed maximal non VD in S and a maximal non local subring of S is established. The relation between dim(R,S) and the number of rings between R and S is given when R is a maximal non VD in S and dim(R, S) is finite. For a maximal non VD R in S such that R R′S S and dim(R, S) is finite, the equality of dim(R,S) and dim(R′S , S) is established.
Maximal non valuative domains
(ARXIV, 2020-09) Kumar, Rahul
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 domain. Maximal non valuative domains have at most four maximal ideals. Various properties of maximal non valuative domains are discussed. Conditions are given under which pseudo-valuation domains and maximal non pseudo-valuation domains are maximal non valuative domains.
Maximal non λ-subrings
(Springer, 2020) Kumar, Rahul
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 R S T , S T is a -extension. We show that a maximal non -subring R of a field has at most two maximal ideals, and exactly two if R is integrally closed in the given field. A determination of when the classical D + M construction is a maximal non -domain is given. A necessary condition is given for decomposable rings to have a field which is a maximal non -subring. If R is a maximal non -subring of a field K, where R is integrally closed in K, then K is the quotient field of R and R is a Prüfer domain. The equivalence of a maximal non -domain and a maximal non valuation subring of a field is established under some conditions. We also discuss the number of overrings, chains of overrings, and the Krull dimension of maximal non -subrings of a field.
Maximal Non ϕ -Chained Rings and Maximal Non Chained Rings
(Springer, 2019-06) Kumar, Rahul
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 domain. A ring R is said to be a maximal non chained (resp., ϕ-chained) subring of S if R is a proper subring of S, R is not a chained (resp., ϕ-chained) ring and every subring of S which contains R properly is a chained (resp., ϕ-chained) ring. We study the properties and characterizations of a maximal non chained (ϕ-chained) subring of a ring. Examples of a maximal non ϕ-chained subring which is not a maximal non chained subring and a maximal non chained subring which is not a maximal non ϕ-chained subring are also given to strengthen the concept.
Maximal non-nonnil-principal ideal rings
(World Scientific, 2025) Kumar, Rahul
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 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]]).
Maximal non-Prüfer and maximal non--Prüfer rings
(Taylor & Francis, 2021-10) Kumar, Rahul
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 concept of maximal non-Prüfer subrings of a field. A proper subring R of a ring S is said to be a maximal non-Prüfer subring of S if R is not a Prüfer ring but every subring of S which contains R properly is a Prüfer ring. A proper subring R of a ring S is said to be maximal non-ϕ-Prüfer subring of S if R is not a ϕ-Prüfer ring but every subring of S which contains R properly is a ϕ-Prüfer ring. We study the properties of maximal non-Prüfer subrings and maximal non-ϕ-Prüfer subrings of a ring in class H. Characterizations of a ring in class H to be a maximal non-Prüfer ring and maximal non-ϕ-Prüfer ring are given. Examples of a maximal non-ϕ-Prüfer subring which is not a maximal non-Prüfer subring and a maximal non-Prüfer subring which is not a maximal non-ϕ-Prüfer subring are also given to strengthen the concept.
Maximal non-pseudovaluation subrings of an integral domain
(Springer, 2024-06) Kumar, Rahul
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 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.
Maximal non-ϕ-pseudo-valuation rings
(World Scientific, 2022) Kumar, Rahul
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 some conditions. The concept of maximal non-PVR is also introduced and it is shown that a maximal non-PVR is either a ϕ-PVR or a maximal non-ϕ-PVR. Examples are also given to show that all the introduced concepts are independent of each other, in general.
A note on -domains and -domains
(The Belgian Mathematical Society, 2020) Kumar, Rahul
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 Δ-domain. We show that if R⊂T is an extension of integral domains such that each proper subring of T containing R is a λ-domain (resp., Δ-domain), then T is a λ-domain (resp., Δ-domain under some conditions). Moreover, the pair (R,T) is a residually algebraic pair. Two new ring theoretic properties, namely λ-property of domains and Δ-property of domains are introduced and studied.
A Note on FMS Modules and FCP Extensions
(Springer, 2022-12) Kumar, Rahul
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 S(+)N⊆S(+)M has FCP. We also discuss FMS modules.
A note on imbedded prime divisors
(Springer, 2020-06) Kumar, Rahul
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
(World Scientific, 2023) Kumar, Rahul
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
(World Scientific, 2023) Kumar, Rahul
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.