Classification of Rings with Genus One Zero-Divisor Graphs

REFERENCES

• Akbari , S. , Maimani , H. R. , Yassemi , S. ( 2003 ). When a zero-divisor graph is planar or complete r-bipartite graph . J. Algebra 270 : 169 – 180 .
   Web of Science ®Google Scholar
• Akbari , S. , Mohammadian , A. ( 2004 ). On the zero-divisor graph of a commutative ring . J. Algebra 274 ( 2 ): 847 – 855 .
   Web of Science ®Google Scholar
• Anderson , D. F. , Livigston , P. S. ( 1999 ). The zero-divisor graph of a commutative ring . J. Algebra 217 : 434 – 447 .
   Web of Science ®Google Scholar
• Anderson , D. F. , Frazier , A. , Lauve , A. , Livingston , P. S. ( 2001 ). The zero-divisor graph of a cummutative ring, II . Deal Theoretic Method in Commutative Algebra ( Columbia, MO, 1999 ). New York : Dekker , pp. 61 – 72 .
   Google Scholar
• Battle , J. , Harary , F. , Kodama , Y. , Youngs , J. W. T. ( 1962 ). Additivity of the genus of a graph . Bull. Amer. Math. Soc. 68 : 565 – 568 .
   Web of Science ®Google Scholar
• Belshoff , R. , Chapman , J. ( 2007 ). Planar zero-divisor graph . J. Algebra 316 ( 1 ): 471 – 480 .
   Web of Science ®Google Scholar
• Corbas , B. ( 1969 ). Rings with few zero divisors . Math. Ann. 181 : 1 – 7 .
   Web of Science ®Google Scholar
• Corbas , B. , Williams , G. D. ( 2000a ). Rings of order p 5 Part I. Nonlocal rings . J. Algebra 231 : 677 - 690 .
   Web of Science ®Google Scholar
• Corbas , B. , Williams , G. D. ( 2000b ). Rings of order p 5 Part II. Local rings . J. Algebra 231 : 691 – 704 .
   Web of Science ®Google Scholar
• Diestel , R. ( 2000 ). Graph Theory, Electronic Edition 2000 . New York : Springer-Verlag .
   Google Scholar
• Krull , W. ( 1924 ). Algebraische theorie der ringe, II . Math. Ann. 91 : 1 – 46 .
   Google Scholar
• Maimani , H. R. , Poumaki , M. R. , Yassemi , S. ( 2006 ). Zero-divisor graph with respect to an ideal . J. Coram. Algebra 34 : 923 – 929 .
   Web of Science ®Google Scholar
• Matsumura , H. ( 1986 ). Commutative Ring Theory . Cambridge : Cambridge University Press .
   Google Scholar
• McDonald , B. R. ( 1974 ). Finite Rings with Identity . New York : Marcel Dekker, Inc .
   Google Scholar
• McKenzie , T. , Overbay , S. ( 2006 ). Book embeddings and zero divisors, Preprint .
   Google Scholar
• Neufeld , E. , Myrvold , W. ( 1997 ). Practical toroidality testing . Eight Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) , pp. 574 – 580 .
   Google Scholar
• Redmond , S. P. ( 2003 ). An ideal-based zero-divisor graph of a commutative ring . Comm. Algebra 31 ( 9 ): 4425 – 4443 .
   Web of Science ®Google Scholar
• Ringel , G. ( 1965 ). Das gescblecht des vollstandingen paaren graphen . Abh. Math. Sem. Univ. Hamburg 28 : 139 – 150 .
   Google Scholar
• Ringel , G. , Youngs , J. W. T. ( 1968 ). Solution of the heawood map-coloring problem . Proc. Nat. Acad. Sei. USA 60 : 438 – 445 .
   PubMed Web of Science ®Google Scholar
• Smith , N. O. ( 2002 ). Planar zero-divisor graphs . Internal. J. Commutative Rings 2 ( 4 ): 177 – 188 .
   Google Scholar
• Wang , H. ( 2006 ). Zero-divisor graphs of genus one . J. Algebra 304 ( 2 ): 666 – 678 .
   Web of Science ®Google Scholar
• White , A. T. ( 1973 ). Graphs, Groups and Surfaces . North-Holland : Amsterdam .
   Google Scholar
• Communicated by I. Swanson.
   Google Scholar