arXiv subjects

Ganesh S. Kadu

Publications and source records attributed to Ganesh S. Kadu.

5 recordsLinked to original sources

arXiv2026

Zero divisors of Gorenstein Rings

Let $R$ be a commutative Artinian ring. We consider two graphs associated to $R$, namely the compressed zero-divisor graph $Γ_E(R)$ and the associate class graph $Γ_A(R)$. Partitioning the vertex set of a zero-divisor graph into its core and its boundary, we count the core vertices that dominate the core. This count is a graph invariant, and we estimate it for $Γ(R)$, $Γ_A(R)$ and $Γ_E(R)$. We prove that the count for $Γ_A(R)$ is bounded below by the count for $Γ_E(R)$, and that the lower bound is attained precisely when $R$ is Gorenstein. As a consequence we obtain that $R$ is Gorenstein if and only if $Γ_A(R)\congΓ_E(R)$ as graphs, the isomorphism being an arbitrary one and not merely the natural compression map. Using the same counting technique we then answer, for Artinian rings, a question of Anderson and LaGrange by showing that $Γ(R)\congΓ_E(R)$ if and only if $R\cong \mathbb Z_2^{\,n}$ for some $n\ge2$, or $R\cong\mathbb Z_4$, or $R\cong\mathbb Z_2[x]/(x^2)$.

math.AC↗

arXiv2025

Enhanced power graph from the power graph of a group

The power graph of a group $G$ is a graph with vertex set $G$, where two distinct vertices $a$ and $b$ are adjacent if one of $a$ and $b$ is a power of the other. Similarly, the enhanced power graph of $G$ is a graph with vertex set $G$, where two distinct vertices are adjacent if they belong to the same cyclic subgroup. In this paper we give a simple algorithm to construct the enhanced power graph from the power graph of a group without the knowledge of the underlying group. This answers a question raised by Peter J. Cameron of constructing enhanced power graph of group $G$ from its power graph. We do this by defining an arithmetical function on finite group $G$ that counts the number of closed twins of a given vertex in the power graph of a group. We compute this function and prove many of its properties. One of the main ingredients of our proofs is the monotonicity of this arithmetical function on the poset of all cyclic subgroups of $G$.

math.GR↗

arXiv2020

On the problem of the finiteness of the compressed zero divisor graphs of Artinian rings

Let R be an Artinian ring and G be the compressed zero-divisor graph associated to R. The question of when the clique number of compressed zero-divisor graphs is finite was raised by J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, and S. Spiroff, in their survey paper entitled On Zero-divisor Graphs. They proved that if length of R is at most four then the clique number is finite. In the length six case they gave an example of a ring where the clique is infinite. In this paper we show that when length of ring is five then the clique number of compressed zero-divisor graph is finite.

math.CO↗

arXiv2019

Bass and Betti Numbers of $A/I^n.$

Let $(A, \m, k)$ be a Gorenstein local ring of dimension $ d\geq 1.$ Let $I$ be an ideal of $A$ with $\htt(I) \geq d-1.$ We prove that the numerical function \[ n \mapsto \ell(\ext_A^i(k, A/I^{n+1}))\] is given by a polynomial of degree $d-1 $ in the case when $ i \geq d+1 $ and $\curv(I^n) > 1$ for all $n \geq 1.$ We prove a similar result for the numerical function \[ n \mapsto \ell(\Tor_i^A(k, A/I^{n+1}))\] under the assumption that $A$ is a \CM ~ local ring. \noindent We note that there are many examples of ideals satisfying the condition $\curv(I^n) > 1,$ for all $ n \geq 1.$ We also consider more general functions $n \mapsto \ell(\Tor_i^A(M, A/I_n)$ for a filtration $\{I_n \}$ of ideals in $A.$ We prove similar results in the case when $M$ is a maximal \CM ~ $A$-module and $\{I_n=\overline{I^n} \}$ is the integral closure filtration, $I$ an $\m$-primary ideal in $A.$

math.AC↗

arXiv2011

Analytic Deviation One Ideals and Test Modules

Let A be a Cohen-Macaulay local ring of dimension d and I an ideal in A. Let M be a finitely generated maximal Cohen-Macaulay A-module. Let I be a locally complete intersection ideal of analytic deviation one and reduction number at most one. We prove that the polynomial given by $length(Tor^{A}_{1}(M,A/I^{n+1}))$ either has degree d-1 or $F_I(M) $ is a free$F(I)-$$module.

math.AC↗