On zero divisor graph of unique product monoid rings over Noetherian reversible ring

Document Type: Research Paper

Authors

Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.

Abstract

 Let $R$ be an associative ring with identity and $Z^*(R)$ be its set of non-zero zero divisors.  The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is the graph whose vertices are the non-zero  zero-divisors of  $R$, and two distinct vertices $r$ and $s$ are adjacent if and only if $rs=0$ or $sr=0$.  In this paper, we bring some results about undirected zero-divisor graph of a monoid ring over reversible right (or left) Noetherian ring $R$. We essentially classify the diameter-structure of this graph and show that $0\leq \mbox{diam}(\Gamma(R))\leq \mbox{diam}(\Gamma(R[M]))\leq 3$. Moreover, we give a characterization for the possible diam$(\Gamma(R))$ and diam$(\Gamma(R[M]))$, when $R$ is a reversible Noetherian ring and $M$ is a u.p.-monoid. Also, we study relations between the girth of $\Gamma(R)$ and that of $\Gamma(R[M])$.

Keywords


[1] S. Akbari and A. Mohammadian, Zero-divisor graphs of non-commutative rings, J.
Algebra 296 (2006), 462-479.
[2] A. Alhevaz and D. Kiani, On zero divisors in skew inverse Laurent series over
noncommutative rings, Comm. Algebra 42(2) (2014), 469-487.
[3] D.D. Anderson and V. Camillo, Semigroups and rings whose zero products commute,
Comm. Algebra 27(6) (1999), 2847-2852.
[4] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring,
J. Algebra 217 (1999), 434-447.
[5] D.F. Anderson and S.B. Mulay, On the diameter and girth of a zero-divisor graph,
J. Pure Appl. Algebra 210 (2007), 543-550.
[6] D.D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra
159 (1993), 500-514.
[7] M. Axtell, J. Coykendall, and J. Stickles, Zero-divisor graphs of polynomials and
power series over commutative rings, Comm. Algebra 33 (2005), 2043-2050.
[8] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208-226.
[9] G.F. Birkenmeier and J.K. Park, Triangular matrix representations of ring exten-
sions, J. Algebra 265 (2003), 457-477.
[10] V. Camillo and P.P. Nielsen, McCoy rings and zero-divisors, J. Pure Appl. Algebra
212 (2008), 599-615.
[11] P.M. Cohn, Reversibe rings, Bull. London Math. Soc. 31 (1999), 641-648.
[12] D.E. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Amer.
Math. Soc. 27(3) (1971), 427-433.
[13] E. Hashemi, McCoy rings relative to a monoid, Comm. Algebra 38 (2010), 1075-
1083.
[14] E. Hashemi and R. Amirjan, Zero divisor graphs of Ore extensions over reversible
rings, submitted.
[15] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative
ring, Trans. Amer. Math. Soc. 115 (1965), 110-130.
[16] G. Hinkle and J.A. Huckaba, The generalized Kronecker function ring and the ring
R(X), J. Reine Angew. Math. 292 (1977), 25-36.
[17] C.Y. Hong, N.K. Kim, Y. Lee, and S.J. Ryu, Rings with Property (A) and their
extensions, J. Algebra 315 (2007), 612-628.
[18] J.A. Huckaba and J.M. Keller, Annihilation of ideals in commutative rings, Paci c
J. Math. 83 (1979), 375-379.
[19] I. Kaplansky, Commutative Rings", University of Chicago Press, Chicago, 1974.
[20] N.K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra 210
(2007), 543-550.
[21] J. Krempaand and D. Niewieczerzal, Rings in which annihilators are ideals and their
application to semigroup rings, Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phys.
25 (1977), 851-856.
[22] T.Y. Lam, A First Course in Noncommutative Rings", Springer-Verlag, 1991.
[23] Z. Liu, Armendariz rings relative to a monoid, Comm. Algebra 33 (2005), 649-661.
[24] T. Lucas, The diameter of a zero divisor graph, J. Algebra 301 (2006), 174-193.
[25] N.H. McCoy, Annihilators in polynomial rings, Amer. Math. Monthly 64 (1957),
28-29.
[26] P.P. Nielsen, Semi-commutativity and the McCoy condition, J. Algebra 298 (2006),
134-141.
[27] J. Okninski, Semigroup Algebras", Marcel Dekker, New York, 1991.
[28] D.S. Passman, The Algebraic Structure of Group Rings", Wiley-Intersceince, New
York, 1977.
[29] Y. Quentel, Sur la compacite du spectre minimal d'un anneau, Bull. Soc. Math.
France 99 (1971), 265-272.
[30] S.P. Redmond, The zero-divisor graph of a non-commutative ring, Int. J. Commut.
Rings 1 (2002), 203-211.
[31] S.P. Redmond, Structure in the zero-divisor graph of a non-commutative ring, Hous-
ton J. Math. 30(2) (2004), 345-355.