Let R be a ring(not necessarily commutative)with identity 1 and let CL(R)be its clean graph.In this paper,we investigate the genus number of the compact Riemann surface in which CL(R)can be embedded and explicitly det...Let R be a ring(not necessarily commutative)with identity 1 and let CL(R)be its clean graph.In this paper,we investigate the genus number of the compact Riemann surface in which CL(R)can be embedded and explicitly determine all commutative rings R(up to isomorphism)such that CL(R)has genus at most two.It is shown that for any Artinian ring R,CL(R)is a projective graph if and only if R is isomorphic to F4.Furthermore,we determine all isomorphism classes of commutative rings whose clean graphs have crosscap two.展开更多
1987年厄尔尼诺的观测表明,当海表温度较高时,温室效应随海表温度增加而增加,其增加率超过海表射出辐射随温度的增加率。由于这种“超温室效应”产生了高反射的卷云,其作用像个恒温器,阻挡住太阳辐射使不能达到海表。这些卷云的调节作...1987年厄尔尼诺的观测表明,当海表温度较高时,温室效应随海表温度增加而增加,其增加率超过海表射出辐射随温度的增加率。由于这种“超温室效应”产生了高反射的卷云,其作用像个恒温器,阻挡住太阳辐射使不能达到海表。这些卷云的调节作用使海表温度只能达到305 K 以下。展开更多
基金supported by Dr.D.S.Kothari Postdoctoral Fellowship Scheme(No.F.4-2/2006(BSR)/MA/17-18/0045)of University Grants Commission,India,awarded to the first authorthe second author was supported by UGC SAP DRS-1(No.F.510/7/DRS-1/2016(SAP-I),University Grants Commission,Government of IndiaThe research of S.Pirzada was supported by SERB-DST research project number CRG/2020/000109.
文摘Let R be a ring(not necessarily commutative)with identity 1 and let CL(R)be its clean graph.In this paper,we investigate the genus number of the compact Riemann surface in which CL(R)can be embedded and explicitly determine all commutative rings R(up to isomorphism)such that CL(R)has genus at most two.It is shown that for any Artinian ring R,CL(R)is a projective graph if and only if R is isomorphic to F4.Furthermore,we determine all isomorphism classes of commutative rings whose clean graphs have crosscap two.
