Narrow Results

Let $R$ be a commutative ring with identity and $I^{\ast }(R),$ the set of all nontrivial proper ideals of $R$. The intersection graph of ideals of $R$, denoted by ${ℐ}(R)$, is a simple undirected graph with vertex set as the set $I^{\ast }(R)$, and, for any two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J\ne (0)$. In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring $R$, ${ℐ}(R)$ is a projective graph if and only if $R\cong R_1\times F_1,$ where $R_1$ is a local principal ideal ring with maximal ideal ${𝔪}_1$ of nilpotency three and $F_1$ is a field. Furthermore, it is shown that for an Artinian ring $R,$$\overline{\gamma }({ℐ}(R))=2$ if and only if $R\cong R_1\times R_2,$ where each $R_i(1\le i\le 2)$ is a local principal ideal ring with maximal ideal ${𝔪}_i\ne (0)$ such that ${𝔪}_i^2=(0).$

Studying algebraic structures via graphs and hypergraphs assigned to them can be of interest. Especially, computing the genus of a graph as a topological index leads to a better understanding of the related algebraic structure. In this direction we apply a hypergraph, namely $3$-zero divisor hypergraph assigned to a commutative ring and study its genus. In this paper, we characterize all finite commutative nonlocal rings $A$ with identity whose ${{\mathscr{H}}}_3(A)$ has genus two. Further, we classify all finite commutative nonlocal rings $A$ whose ${{\mathscr{H}}}_3(A)$ has crosscap two. Moreover, we provide a MATLAB code for calculating $3$-zero-divisor of ${\mathbb{Z}}_n$ and the hyperedge of $3$-zero-divisor hypergraph of ${\mathbb{Z}}_n$.

The apparent contour of a surface in $3$-space can be investigated in terms of singularity theory. We show the precise bifurcation diagrams of the apparent contours of generic crosscap surfaces with respect to orthogonal projections. Especially, our bifurcation diagrams contain also the information of the projected images of the singular sets of crosscap surfaces.

Let $R$ be a ring (not necessarily commutative) with identity 1 and let $\mathcal{CL}(R)$ be its clean graph. In this paper, we investigate the genus number of the compact Riemann surface in which $\mathcal{CL}(R)$ can be embedded and explicitly determine all commutative rings $R$ (up to isomorphism) such that $\mathcal{CL}(R)$ has genus at most two. It is shown that for any Artinian ring $R$, $\mathcal{CL}(R)$ is a projective graph if and only if $R$ is isomorphic to ${\mathbb{F}}_4$. Furthermore, we determine all isomorphism classes of commutative rings whose clean graphs have crosscap two.

Let $R$ be a commutative ring with identity. We consider a simple graph associated with $R$, denoted by ${\mathrm{\Omega }}_R^{\ast }$, whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent whenever $J\text{Ann}(I)=(0)$ or $I\text{Ann}(J)=(0)$. In this paper, we characterize the commutative Artinian non-local ring $R$ for which ${\mathrm{\Omega }}_R^{\ast }$ has genus one and crosscap one.

Let $R$ be a commutative ring with identity. The cozero-divisor graph of $R$, denoted by ${\mathrm{\Gamma }}^{{}^{\prime }}(R)$, is a graph whose vertex set is $W^{\ast }(R)$, the set of all non-zero and non-unit elements of $R$. Two distinct vertices $a$ and $b$ in $W^{\ast }(R)$ are adjacent if and only if $a\notin Rb$ and $b\notin Ra$. The Crosscap of a graph $G$, denoted by $\overline{\gamma }(G)$, is the minimum integer $k$ such that the graph can be embedded in the non-orientable surface ${{𝒩}}_k$. The planar graph is called $k$-outerplanar if removing all the vertices incident on the outer face yields a $(k-1)$-outerplanar. The Outerplanarity index of a graph $G$ is the smallest $k$ such that $G$ is $k$-outerplanar. In this paper, we characterize the class of rings $R$ (up to isomorphism) for which $0<\overline{\gamma }({\mathrm{\Gamma }}^{{}^{\prime }}(R))\le 2$. Further we characterize all finite rings $R$ (up to isomorphism) for which ${\mathrm{\Gamma }}^{{}^{\prime }}(R)$ has an outerplanarity index two.

Let $R$ be a commutative ring with identity and $Z{(R)}^{\ast }$ be the set of all nonzero zero-divisors of $R$. The annihilator graph of commutative ring $R$ is the simple undirected graph AG($R$) with vertices $Z{(R)}^{\ast },$ and two distinct vertices $x$ and $y$ are adjacent if and only if $\mathrm{\text{ann}}(xy)\ne \mathrm{\text{ann}}(x)\cup \mathrm{\text{ann}}(y).$ The essential graph of $R$ is defined as the graph $\mathrm{\text{EG}}(R)$ with the vertex set $Z{(R)}^{\ast }=Z(R)\setminus \{0\},$ and two distinct vertices $x$ and $y$ are adjacent if and only if ${\mathrm{\text{ann}}}_R(xy)$ is an essential ideal. In this paper, we classify all finite commutative rings with identity whose annihilator graph and essential graph have crosscap two.

Let $R$ be a finite commutative ring with identity, $I$ be an ideal of $R$ and $J(R)$ denotes the Jacobson radical of $R$. The ideal-based zero-divisor graph ${\mathrm{\Gamma }}_I(R)$ of $R$ is a graph with vertex set $V({\mathrm{\Gamma }}_I(R))=\{x\in R-I:xy\in I,$ for some $y\in R-I\}$ in which distinct vertices $x$ and $y$ are adjacent if and only if $xy\in I$. In this paper, we determine the diameter, girth of ${\mathrm{\Gamma }}_{J(R)}(R)$. Specifically, we classify all finite commutative nonlocal rings for which ${\mathrm{\Gamma }}_{J(R)}(R)$ is perfect. Furthermore, we discuss about the planarity, outerplanarity, genus and crosscap of ${\mathrm{\Gamma }}_{J(R)}(R)$ and characterize all of them.