Let R be a commutative ring and U(R)the multiplicative group of unit elements of R.In 2012,Khashyarmanesh et al.defined the generalized unit and unitary Cayley graph,T(R,G,S),corresponding to a multiplicative subgroup...Let R be a commutative ring and U(R)the multiplicative group of unit elements of R.In 2012,Khashyarmanesh et al.defined the generalized unit and unitary Cayley graph,T(R,G,S),corresponding to a multiplicative subgroup G of U(R)and a nonempty subset S of G with S^(-1)={s^(-1)|s∈S}■S,asthegraphwithvertexsetR and two distinct vertices x and y being adjacent if and only if there exists s∈S such that x+sy∈G.In this paper,we characterize all Artinian rings R for which T(R,U(R),S)is projective.This leads us to determine all Artinian rings whose unit graphs,unitary Cayley graphs and co-maximal graphs are projective.In addition,we prove that for an Artinian ring R for which T(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 for which T(R,U(R),S)has nonorientable genus k is finite.展开更多
文摘Let R be a commutative ring and U(R)the multiplicative group of unit elements of R.In 2012,Khashyarmanesh et al.defined the generalized unit and unitary Cayley graph,T(R,G,S),corresponding to a multiplicative subgroup G of U(R)and a nonempty subset S of G with S^(-1)={s^(-1)|s∈S}■S,asthegraphwithvertexsetR and two distinct vertices x and y being adjacent if and only if there exists s∈S such that x+sy∈G.In this paper,we characterize all Artinian rings R for which T(R,U(R),S)is projective.This leads us to determine all Artinian rings whose unit graphs,unitary Cayley graphs and co-maximal graphs are projective.In addition,we prove that for an Artinian ring R for which T(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 for which T(R,U(R),S)has nonorientable genus k is finite.
