Abstract
The graph, whose vertices are the elements of a ring \(R\) and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(xy=0\), is the Beck graph. A graph \((V, E)\) is said to be a split graph if \(V\) is the disjoint union of two sets \(K\) and \(S\) where \(K\) induces a complete subgraph and \(S\) is an independent set. In this paper we are concerned with rings whose Beck graph is split. We call such rings split rings. First, we find all reduced split rings. For a general ring we obtain some partial results.
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References
Anderson, D.F., Livingston, P.S.: The zero-divisor graph of a commutative ring. J. Algebra 217, 434–447 (1999)
Atiyah, M.F., MacDonald, I.G.: Introduction to Commutative Algebra. Addison-Wesely, Reading (1969)
Beck, I.: Coloring of commutative rings. J. Algebra 116, 208–226 (1988)
Ganesan, N.: Properties of rings with a finite number of zero-divisors. Math. Ann. 157, 215–218 (1964)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, San Diego (1980)
Jinnah, M.I., Mathew, S.C.: When is the comaximal graph split? Commun. Algebra 40(7), 2400–2404 (2012)
Parthasarathy, K.R.: Basic Graph Theory. Tata McGraw Hill Publications Company Ltd, New Delhi (1994)
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Jinnah, M.I., Mathew, S.C. On rings whose Beck graph is split. Beitr Algebra Geom 56, 379–385 (2015). https://doi.org/10.1007/s13366-014-0222-6
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DOI: https://doi.org/10.1007/s13366-014-0222-6