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Shiv Datt Kumar

Publications and source records attributed to Shiv Datt Kumar.

13 recordsLinked to original sources
arXiv2025

On S-Integral Domains and S-Version of Krull Intersection Theorem

Let $S\subseteq R$ be a multiplicatively closed subset of a ring $R$. We extend several results on integral domains to their $S$-versions and establish the $S$-version of Krull intersection theorem. We also show that if $R$ is an $S$-field, then the localization of $R$ with respect to $S$ is a $\phi(S)$-field, where $\phi(S)=\left \{\dfrac{s}{1}| \ s\in S\right \}$ is a multiplicatively closed subset of $S^{-1}R$, and prove the converse under the condition of finiteness of $S$. As a consequence, we show that every finite $S$-integral domain is an $S$-field. Also, we provide several examples to illustrate the significance of our findings.

math.AC↗
arXiv2025

On S-J-Noetherian Rings

Let $R$ be a commutative ring with identity, $S\subseteq R$ be a multiplicative set and $J$ be an ideal of $R$. In this paper, we introduce the concept of $S$-$J$-Noetherian rings, which generalizes both $J$-Noetherian rings and $S$-Noetherian rings. We study several properties and charaterizations of this new class of rings. For instance, we prove Cohen's-type theorem for $S$-$J$-Noetherian rings. Among other results, we establish the existence of $S$-primary decomposition in $S$-$J$-Noetherian rings as a generalization of classical Lasker-Noether theorem.

math.AC↗
arXiv2025

Structural Analysis of Commutative S-Reduced Rings

Let $R$ be a commutative ring with identity, $S \subseteq R$ be a multiplicative set. In this paper, we establish that the intersection of all $S$-prime ideals in an $S$-reduced ring is $S$-zero. Also, we show that an $S$-Artinian reduced ring is isomorphic to the finite direct product of fields. Furthermore, we provide an example of an $S$-reduced ring which is a uniformly-$S$-Armendariz ring (in short, $u$-$S$-Armendariz$)$ ring. Additionally, we prove that the class of uniformly-$S$-reduced rings (in short, $u$-$S$-reduced rings) belongs to the class of $u$-$S$-Armendariz rings. Among other results, we establish the relationship between $S$-reduced rings and $S$-strongly Hopfian rings. Finally, we prove the structure theorem for $S$-reduced rings.

math.AC↗
arXiv2025

Nonnil-S-Laskerian Rings

In this paper, we introduce the concept of nonnil-S-Laskerian rings, which generalize both nonnil-Laskerian rings and S-Laskerian rings. A ring R is said to be nonnil-S-Laskerian if every nonnil ideal I (disjoint from S) of R is S-decomposable. As a main result, we prove that the class of nonnil-S-Noetherian rings belongs to the class of nonnil-S-Laskerian rings. Also, we prove that a nonnil-S-Laskerian ring has S-Noetherian spectrum under a mild condition. Among other results, we prove that if the power series ring R[[X]] is nonnil-S-Laskerian with S-decomposable nilradical, then R is S-laskerian and satisfies the S-SFT property.

math.AC↗
arXiv2025

Relative Lie central extension and Schur multiplier of pairs of multiplicative Lie algebras

In this paper, we introduce the concept of relative Lie central extension for pair of multiplicative Lie algebras. Then, we discuss the concept of isoclinism for relative Lie central extensions and prove some related results. We also define the Frattini subalgebra for multiplicative Lie algebras and discuss its properties, finally the Schur multiplier for pair of multiplicative Lie algebras is introduced and under certain conditions prove the existence of multiplicative covering pair.

math.GR↗
arXiv2025

Correspondence, Wells and Hochschild-Serre sequences for nonabelian extensions of multiplicative Lie algebras

For nonabelian $2^{\mathrm{nd}}$-cohomology of multiplicative Lie algebras, we properly generalize from the group case three classic results. We prove a Correspondence theorem which compares $2^{\mathrm{nd}}$-cohomology associated to a realized abstract kernel to the abelian $2^{\mathrm{nd}}$-cohomology group over the algebraic center. For arbitrary extensions, we prove a Wells's Theorem characterizing ideal-preserving automorphisms and establish the 1-dimensional Lyndon-Hochschild-Serre exact sequence. Several previously established results are recovered when restricted to extensions with group-abelian or Lie-trivial ideals.

math.RA↗
arXiv2024

Tensor square and isoclinic extensions of multiplicative Lie algebras

In this paper, we discuss the capable and isoclinic properties of the tensor square in the context of multiplicative Lie algebras. We also developed the concept of isoclinic extensions and proved several results for multiplicative Lie algebras. Consequently, we demonstrate that covers of a multiplicative Lie algebra are mutually isoclinic.

math.GR↗
arXiv2024

Automorphisms of Multiplicative Lie algebra Extensions

In this paper, we discuss the inducibility problem for automorphisms of multiplicative Lie algebra extensions and show that obstruction to the inducibility of pairs lies in the second cohomology group of multiplicative Lie algebras. We also establish the Wells type exact sequence for multiplicative Lie algebras, which relates automorphism groups with the second cohomology group of multiplicative Lie algebras.

math.RA↗
arXiv2024

Sixteen-dimensional Sedenion-like Associative Algebra

In this article, we construct a $16$-dimensional sedenion-like associative algebra, which is an even subalgebra of $2^5$-dimensional Clifford algebra $Cl_{5,0}$. We define the norm on sedenion-like algebra and show that its sixteen-dimensional elements preserves the norm relation $\lVert ST \rVert=\lVert S \rVert \lVert T \rVert$ under the condition $S_rS_d^\dagger + S_r^\dagger S_d=0$, where $S_r,~S_d$ denote the real and dual part of an octonion-like number $S$ respectively and $S^\dagger$ is the transpose of $S$. The elements of this sedenion-like algebra can be written as dual octonion like numbers called split bioctonion-like algebra and $S S^\dagger$ is commutative [i.e. $S S^\dagger=S^\dagger S $ and $(S S^\dagger) T=T(S S^\dagger )$], for any two octonion-like/sedenion-like numbers $S$ and $T$. We define the operations coproduct $\bigtriangleup$, counit $ε$ and antipode $S$ on octonion-like/sedenion-like algebra to construct the Hopf algebra structure on it. We also show that $8$-dimensional octonion-like associative seminormed division algebra is a $\mathbb{Z}_2^4/2$-graded quasialgebra and $16$ dimensional sedenion-like algebra is a $\mathbb{Z}_2^5/2$-graded quasialgebra.

math.AC↗
arXiv2024

A Study of S-Primary Decompositions

Let $R$ be a commutative ring with identity and $S \subseteq R$ be a multiplicative set. An ideal $Q$ of $R$ (disjoint from $S$) is said to be $S$-primary if there exists an $s\in S$ such that for all $x,y\in R$ with $xy\in Q$, we have $sx\in Q$ or $sy\in rad(Q)$. Also, we say that an ideal of $R$ is $S$-primary decomposable or has an $S$-primary decomposition if it can be written as finite intersection of $S$-primary ideals. In this paper, first we provide an example of $S$-Noetherian ring in which an ideal does not have a primary decomposition. Then our main aim of this paper is to establish the existence and uniqueness of $S$-primary decomposition in $S$-Noetherian rings as an extension of a historical theorem of Lasker-Noether.

math.AC↗
arXiv2011

Existence of a Unique group of finite order

Let $n$ be a positive integer. Then cyclic group $Z_n$ of order $n$ is the only group of order $n$ iff g.c.d. $(n,ϕ(n))=1$, where $ϕ$ denotes the Euler-phi function. In this article we have given another proof of this result using the knowledge of semi direct product and induction.

math.HO↗