arXiv subjects
Publications and source records attributed to R. Nikandish.
We construct a family of finite local rings whose annihilating-ideal graphs are naturally described by orthogonality of subspaces of $\mathbb{F}_2^n$. For $n=4$ we determine the clique and chromatic numbers exactly and obtain \[ ω(\AG(R_4))=5<6=χ(\AG(R_4)). \] Thus $\AG(R_4)$ is not weakly perfect, and the Behboodi--Rakeei conjecture fails for non-reduced commutative rings.
The inclusion ideal graph of a commutative unitary ring $R$ is the (undirected) graph $In(R)$ whose vertices all non-trivial ideals of $R$ and two distinct vertices are adjacent if and only if one of them is a proper subset of the other one. In this paper, the metric dimension of $In(R)$ is discussed. Moreover, the structure of the resolving graph of $In(R)$ is characterized and as an application, we compute the strong metric dimension of $In(R)$.
Let $G = (V, E)$ be a graph with the vertex set $V (G)$ and edge set $E(G)$. The Sombor index of $G$, $SO(G)$, is defined as $\sum_{uv\in E(G)} \sqrt{deg(u)^2 + deg(v)^2}$, where $deg(u)$ is the degree of vertex $u$ in $V (G)$. The clean graph of a ring R, denoted by $Cl(R)$, is a graph with vertex set $\{(e, u) : e \in Id(R), u \in U(R)\}$ and two distinct vertices $(e, u)$ and$(f, v)$ are adjacent if and only if $ef = 0$ or $uv = 1$ ($Id(R)$ and $U(R)$ are the sets of idempotents and unit elements of R, respectively). The induced subgraph on $\{(e, u) : e \in Id^{*}(R), u \in U(R)\}$ is denoted by $Cl_2(R)$. In this paper, $SO(Cl2(\mathbb{Z}_n))$, for different values of the positive integer $n$, is investigated.
The intersection graph of ideals associated with a commutative unitary ring $R$ is the graph $G(R)$ whose vertices all non-trivial ideals of $R$ and there exists an edge between distinct vertices if and only if the intersection of them is non-zero. In this paper, the structure of the resolving graph of $G(R)$ is characterized and as an application, we evaluate the strong metric dimension of $G(R)$.
The prime ideal sum graph of a commutative unital ring $R$, denoted by $PIS(R)$, is an undirect and simple graph whose vertices are non-trivial ideals of $R$ and there exists and edge between to distinct vertices if and only if their sum is a prime ideal of $R$. In this paper, the metric dimension of $PIS(R)$ is discussed and some formulae for this parameter in $PIS(R)$ are given.
Let $R$ be a commutative ring with identity. The co-maximal ideal graph of $R$, denoted by $Γ(R)$, is a simple graph whose vertices are proper ideals of $R$ which are not contained in the Jacobson radical of $R$ and two distinct vertices $I, J$ are adjacent if and only if $I+J=R$. In this paper, we use Gallai$^{^,}$s Theorem and the concept of strong resolving graph to compute the strong metric dimension for co-maximal ideal graphs of commutative rings. Explicit formulae for the strong metric dimension, depending on whether the ring is reduced or not, are established.
A famous result due to L. S. Levy provides a classification of all finitely generated indecomposable modules over Dedekind-like rings. This motivates us to outline an approach to the classification of indecomposable pseudo-absorbing primary multiplication modules with finite-dimensional top over certain kinds of pullback rings. In this paper, we give a complete classification, up to isomorphism, of all indecomposable pseudo-absorbing primary multiplication modules with finite-dimensional top over a pullback of two valuation domains with the same residue field. We also find a connection between pseudo-absorbing primary multiplication modules and pure-injective modules over such domains.
An $R$-module $M$ is called virtually uniserial if for every finitely generated submodule $0 \neq K \subseteq M$, $K/$Rad$(K)$ is virtually simple. In this paper, we generalize virtually uniserial modules by dropping the virtually simple condition and replacing it by the cyclic uniform condition. An $R$-module $M$ is called cyclic-uniform uniserial if $K/$Rad$(K)$ is cyclic and uniform, for every finitely generated submodule $0 \neq K \subseteq M$. Also, $M$ is said to be cyclic-uniform serial if it is a direct sum of cyclic-uniform uniserial modules. Several properties of cyclic-uniform (uni)serial modules and rings are given. Moreover, the structure of Noetherian left cyclic-uniform uniserial rings are characterized. Finally, we study rings $R$ have the property that every finitely generated $R$-module is cyclic-uniform serial.
A famous result due to I. M. Isaacs states that if a commutative ring $R$ has the property that every prime ideal is principal, then every ideal of $R$ is principal. This motivates ring theorists to study commutative rings for which every ideal is a direct sum of cyclically presented modules. In this paper, we study commutative rings whose ideals are direct sum of cyclically presented modules.
The essential annihilating-ideal graph $\mathcal{EG}(R)$ of a commutative unital ring $R$ is a simple graph whose vertices are non-zero ideals of $R$ with non-zero annihilator and there exists an edge between two distinct vertices $I,J$ if and only if $Ann(IJ)$ has a non-zero intersection with any non-zero ideal of $R$. In this paper, we show that $\mathcal{EG}(R)$ is weakly perfect, if $R$ is Noetherian and an explicit formula for the clique number of $\mathcal{EG}(R)$ is given. Moreover, the structures of all rings whose essential annihilating-ideal graphs have chromatic number $2$ are fully determined. Among other results, twin-free clique number and edge chromatic number of $\mathcal{EG}(R)$ are examined.
In this paper, the notions of integral closure of hyperrings and hyperideals in a Krasner hyperring $(R, +, \cdot)$ are defined and some basics properties of them are studied. We define also the notion of hypervaluation hyperideals and then a relations between hypervaluations, integral closure of hyperideals and primary hyperideals are studied. In fact it is shown that the integral closure of a hyperideal is determined by the hypervaluation Krasner hyperrings.
Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of weakly $1$-absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing prime if for all nonunit elements $a,b,c \in R$ such that $0\neq abc \in I$, then either $ab \in I$ or $c \in I$. A number of results concerning weakly $1$-absorbing prime ideals and examples of weakly $1$-absorbing prime ideals are given. It is proved that if $I$ is a weakly $1$-absorbing prime ideal of a ring $R$ and $0 \neq I_1I_2I_3 \subseteq I$ for some ideals $I_1, I_2, I_3$ of $R$ such that $I$ is free triple-zero with respect to $I_1I_2I_3$, then $ I_1I_2 \subseteq I$ or $I_3\subseteq I$. Among other things, it is shown that if $I$ is a weakly $1$-absorbing prime ideal of $R$ that is not $1$-absorbing prime, then $I^3 = 0$. Moreover, weakly $1$-absorbing prime ideals of PID's and Dedekind domains are characterized. Finally, we investigate commutative rings with the property that all proper ideals are weakly $1$-absorbing primes.
Let $R$ be a commutative ring with non-zero identity. The cozero-divisor graph of $R$, denoted by $Γ^{\prime}(R)$, is a graph with vertices in $W^*(R)$, which is the set of all non-zero and non-unit elements of $R$, and two distinct vertices $a$ and $b$ in $W^*(R)$ are adjacent if and only if $a\not\in Rb$ and $b\not\in Ra$. In this paper, we show that the cozero-divisor graph of a von Neumann regular ring with finite clique number is not only weakly perfect but also perfect. Also, an explicit formula for the clique number is given.
Let $R$ be a commutative ring with identity and ${\rm Nil}(R)$ be the set of nilpotent elements of $R$. The nil-graph of ideals of $R$ is defined as the graph $\mathbb{AG}_N(R)$ whose vertex set is $\{I:\ (0)\neq I\lhd R$ and there exists a non-trivial ideal $J$ such that $IJ\subseteq {\rm Nil}(R)\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ\subseteq {\rm Nil}(R)$. Here, we study conditions under which $\mathbb{AG}_N(R)$ is complete or bipartite. Also, the independence number of $\mathbb{AG}_N(R)$ is determined, where $R$ is a reduced ring. Finally, we classify Artinian rings whose nil-graphs of ideals have genus at most one.
A graph is called weakly perfect if its vertex chromatic number equals its clique number. Let $R$ be a ring and $I(R)^*$ be the set of all left proper non-trivial ideals of $R$. The intersection graph of ideals of $R$, denoted by $G(R)$, is a graph with the vertex set $I(R)^*$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J\neq 0$. In this paper, it is shown that $G(\mathbb{Z}_n)$, for every positive integer $n$, is a weakly perfect graph. Also, for some values of $n$, we give an explicit formula for the vertex chromatic number of $G(\mathbb{Z}_n)$. Furthermore, it is proved that the edge chromatic number of $G(\mathbb{Z}_n)$ is equal to the maximum degree of $G(\mathbb{Z}_n)$ unless either $G(\mathbb{Z}_n)$ is a null graph with two vertices or a complete graph of odd order.