A class of zero divisor rings in which every graph is precisely the union of a complete graph and a complete bipartite graph

Syed Khalid Nauman; Basmah H. Shafee

Open Mathematics (2015)

  • Volume: 13, Issue: 1
  • ISSN: 2391-5455

Abstract

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Recently, an interest is developed in estimating genus of the zero-divisor graph of a ring. In this note we investigate genera of graphs of a class of zero-divisor rings (a ring in which every element is a zero divisor). We call a ring R to be right absorbing if for a; b in R, ab is not 0, then ab D a. We first show that right absorbing rings are generalized right Klein 4-rings of characteristic two and that these are non-commutative zero-divisor local rings. The zero-divisor graph of such a ring is proved to be precisely the union of a complete graph and a complete bipartite graph. Finally, we have estimated lower and upper bounds of the genus of such a ring.

How to cite

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Syed Khalid Nauman, and Basmah H. Shafee. "A class of zero divisor rings in which every graph is precisely the union of a complete graph and a complete bipartite graph." Open Mathematics 13.1 (2015): null. <http://eudml.org/doc/276008>.

@article{SyedKhalidNauman2015,
abstract = {Recently, an interest is developed in estimating genus of the zero-divisor graph of a ring. In this note we investigate genera of graphs of a class of zero-divisor rings (a ring in which every element is a zero divisor). We call a ring R to be right absorbing if for a; b in R, ab is not 0, then ab D a. We first show that right absorbing rings are generalized right Klein 4-rings of characteristic two and that these are non-commutative zero-divisor local rings. The zero-divisor graph of such a ring is proved to be precisely the union of a complete graph and a complete bipartite graph. Finally, we have estimated lower and upper bounds of the genus of such a ring.},
author = {Syed Khalid Nauman, Basmah H. Shafee},
journal = {Open Mathematics},
keywords = {Right (left) absorbing rings; Klein 4-rings; Zero-divisor (di)graphs; Genus of a ring},
language = {eng},
number = {1},
pages = {null},
title = {A class of zero divisor rings in which every graph is precisely the union of a complete graph and a complete bipartite graph},
url = {http://eudml.org/doc/276008},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Syed Khalid Nauman
AU - Basmah H. Shafee
TI - A class of zero divisor rings in which every graph is precisely the union of a complete graph and a complete bipartite graph
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - null
AB - Recently, an interest is developed in estimating genus of the zero-divisor graph of a ring. In this note we investigate genera of graphs of a class of zero-divisor rings (a ring in which every element is a zero divisor). We call a ring R to be right absorbing if for a; b in R, ab is not 0, then ab D a. We first show that right absorbing rings are generalized right Klein 4-rings of characteristic two and that these are non-commutative zero-divisor local rings. The zero-divisor graph of such a ring is proved to be precisely the union of a complete graph and a complete bipartite graph. Finally, we have estimated lower and upper bounds of the genus of such a ring.
LA - eng
KW - Right (left) absorbing rings; Klein 4-rings; Zero-divisor (di)graphs; Genus of a ring
UR - http://eudml.org/doc/276008
ER -

References

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  10. [10] Marks G., A taxonomy of 2-primal rings, J. Algebra, 266, (2003), 494-520. Zbl1045.16001
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  12. [12] Shafee B.H., Nauman S.K., On extensions of right symmetric rings without identity, Adv. Pure Math., (2014) 4, 665-673. 
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