论文标题
在通用单元的不可取向属上，统一环的统一卡利图
On the nonorientable genus of the generalized unit and unitary Cayley graphs of a commutative ring
论文作者
论文摘要
令$ r $为可交换戒指，让$ u（r）$为$ r $的单位元素的乘法组。 2012年，Khashyarmanesh等人。定义的广义单元和统一的Cayley图，$γ（r，g，s）$，对应于$ u（r）$的乘法亚组$ g $ of $ u（r）$和$ s^{ - 1} = \ s^{ - 1} = \ s^{s^{-1}} $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $顶点$ x $和$ y $相邻，并且仅当存在$ s \ in s $中，以便$ x+sy \ in g $。在本文中，我们表征了所有Artinian Rings $ r $，其$γ（R，U（R），S）$都是投影的。这导致确定所有的Artinian环，其单位图，统一的Cayley Garphs和联合最大图形均具有投影。另外，我们证明，对于Artinian Ring $ r $，其$γ（r，u（r），s）$具有有限的不可取向属，$ r $必须是有限的环。最后，事实证明，对于给定的正整数$ k $，有限环$ r $的数量$γ（r，u（r），s）$具有不可方向的属$ k $是有限的。
Let $R$ be a commutative ring and let $U(R)$ be multiplicative group of unit elements of $R$. In 2012, Khashyarmanesh et al. defined generalized unit and unitary Cayley graph, $Γ(R, G, S)$, corresponding to a multiplicative subgroup $G$ of $U(R)$ and a non-empty subset $S$ of $G$ with $S^{-1}=\{s^{-1} \mid s\in S\}\subseteq S$, as the graph with vertex set $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if there exists $s\in S$ such that $x+sy \in G$. In this paper, we characterize all Artinian rings $R$ whose $Γ(R,U(R), S)$ is projective. This leads to determine all Artinian rings whose unit graphs, unitary Cayley garphs and co-maximal graphs are projective. Also, we prove that for an Artinian ring $R$ whose $Γ(R, U(R), S)$ has finite nonorientable genus, $R$ must be a finite ring. Finally, it is proved that for a given positive integer $k$, the number of finite rings $R$ whose $Γ(R, U(R), S)$ has nonorientable genus $k$ is finite.
