Upper ideal relation graphs associated to rings

Abstract

Let $R$ be a ring with unity. The upper ideal relation graph $Γ_U(R)$ of the ring $R$ is a simple undirected graph whose vertex set is the set of all non-unit elements of $R$ and two distinct vertices $x, y$ are adjacent if and only if there exists a non-unit element $z \in R$ such that the ideals $(x)$ and $(y)$ contained in the ideal $(z)$. In this article, we classify all the non-local finite commutative rings whose upper ideal relation graphs are split graphs, threshold graphs and cographs, respectively. In order to study topological properties of $Γ_U(R)$, we determine all the non-local finite commutative rings $R$ whose upper ideal relation graph has genus at most $2$. Further, we precisely characterize all the non-local finite commutative rings for which the crosscap of $Γ_U(R)$ is either $1$ or $2$.

Figures (6)

• Figure 1: Planar drawing of $\Gamma_U(\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2)$
• Figure 2: Planar drawing of (a) $\Gamma_U(\mathbb{Z}_2 \times \mathbb{Z}_4)$ and (b) $\Gamma_U(\mathbb{Z}_2 \times \frac{\mathbb{Z}_2[x]}{(x^2)})$
• Figure 3: Embedding of (a) $\Gamma_U(\mathbb{Z}_3 \times \mathbb{Z}_4)$ and (b) $\Gamma_U(\mathbb{Z}_3 \times \frac{\mathbb{Z}_2[x]}{(x^2)})$ in $\mathbb{S}_1$
• Figure 4: Embedding of $\Gamma_U(\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3)$ in $\mathbb{S}_2$
• Figure 5: Embedding of $\Gamma_U(\mathbb{F}_4 \times \mathbb{Z}_5)$ in $\mathbb{N}_1$

Theorems & Definitions (42)

• Lemma 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 2.4
• Lemma 2.5
• Lemma 2.6
• Proposition 2.7
• Lemma 2.8
• Definition 2.9
• Lemma 2.10