On the nonorientable genus of the generalized unit and unitary Cayley graphs of a commutative ring

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, $\Gamma(R, G, S)$, 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 $\Gamma(R,U(R), S)$ 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 $\Gamma(R, U(R), S)$ 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 $\Gamma(R, U(R), S)$ has nonorientable genus $k$ is finite.Comment: To appear in Algebra Colloquiu

Some factorization properties of idealization in commutative rings with zero divisors

We study some factorization properties of the idealization \idealztn{R}{M} of a module $M$ in a commutative ring $R$ which is not necessarily a domain. We show that \idealztn{R}{M} is ACCP if and only if $R$ is ACCP and $M$ satisfies ACC on its cyclic submodules, provided that $M$ is finitely generated. We give an example to show that the BF property is not necessarily preserved in idealization, and give some conditions under which \idealztn{R}{M} is a BFR. We also characterize the idealization rings which are UFRs