×

On diameter of the zero-divisor and the compressed zero-divisor graphs of skew Laurent polynomial rings. (English) Zbl 1465.16024

Let \(R\) be an associative ring with nonzero identity. A ring \(R\) is called reversible if \(ab = 0\) implies \(ba = 0\), for \(a,b \in R\). \(R\) is called \(\alpha\)-compatible if for each \(a, b \in R, ab = 0 \iff a\alpha(b) = 0\). The zero-divisor graph \(\Gamma (R)\) of \(R\) is the (undirected) graph with vertices the nonzero zero-divisors of \(R\), and distinct vertices \(a\) and \(b\) are adjacent if and only if \(ab = 0\) or \(ba = 0\). Let \(r_R(a)\) and \(l_R(a)\) be the set of all right annihilators and the set of all left annihilator of an element \(a \in R\), respectively, and let \(\mathrm{ann}_R(a) = l_R(a) \cup r_R(a)\). The relation on \(R\) given by \(a \sim b\) if and only if \(\mathrm{ann}_R(a) = \mathrm{ann}_R(b)\) is an equivalence relation. The graph of equivalence classes of zero-divisors of a ring \(R\) which is called the compressed zero-divisor graph of \(R\) and denoted by \(\Gamma _E(R)\). \(\Gamma _E(R)\) is the (undirected) graph with vertices the equivalence classes induced by \(\sim\) other than the classes \( [0]_R\) and \([1]_R\), and distinct vertices \([a]_R\) and \([b]_R\) are adjacent if and only if \(ab = 0\) or \(ba = 0\). The diameter of a graph \(\Gamma\) is defined as follows: diam(\(\Gamma \)) = sup \(\{d(a, b) \mid a \text{ and }b \text{ are distinct vertices of } \Gamma \}\). In this paper, the authors study the diameter of zero-divisor and the compressed zero-divisor graph of skew Laurent polynomial rings over noncommutative rings \(R[x, x^{-1}; \alpha]\). They give a complete characterization of the possible diameters of \(\Gamma (R[x, x^{-1}; \alpha])\) and \(\Gamma _E(R[x, x^{-1}; \alpha])\), where the base ring \(R\) is reversible and also has the \(\alpha\)-compatible property.

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
05C12 Distance in graphs
16U80 Generalizations of commutativity (associative rings and algebras)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alhevaz, A. and Kiani, D., On zero divisors in skew inverse Laurent series over noncommutative rings, Comm. Algebra42(2) (2014) 469-487. · Zbl 1298.16023
[2] Alhevaz, A. and Kiani, D., McCoy property of skew Laurent polynomials and power series rings, J. Algebra Appl.13(2) (2014), Article ID:1350083, 23 pp. · Zbl 1295.16026
[3] Anderson, D. F. and LaGrange, J. D., Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph, J. Pure Appl. Algebra216 (2012) 1626-1636. · Zbl 1254.13003
[4] Anderson, D. F. and LaGrange, J. D., Abians poset and the ordered monoid of annihilator classes in a reduced commutative ring, J. Algebra Appl.13(8) (2014), Article ID: 1450070, 18 pp. · Zbl 1317.06020
[5] Anderson, D. F. and LaGrange, J. D., The semilattice of annihilator classes in a reduced commutative ring, Comm. Algebra43 (2015) 29-42. · Zbl 1318.13010
[6] Anderson, D. F. and LaGrange, J. D., Some remarks on the compressed zero-divisor graph, J. Algebra447 (2016) 297-321. · Zbl 1330.13003
[7] Anderson, D. F. and Livingston, P. S., The zero-divisor graph of a commutative ring, J. Algebra217 (1999) 434-447. · Zbl 0941.05062
[8] Anderson, D. F. and Mulay, S. B., On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra210 (2007) 543-550. · Zbl 1119.13005
[9] Akbari, S. and Mohammadian, A., On the zero-divisor graph of a commutative ring, J. Algebra274 (2004) 847-855. · Zbl 1085.13011
[10] Axtell, M., Coykendall, J. and Stickles, J., Zero-divisor graphs of polynomials and power series over commutative rings, Comm. Algebra6 (2005) 2043-2050. · Zbl 1088.13006
[11] Azarpanah, F., Karamzadeh, O. A. S. and Aliabad, A. Rezai, On ideals consisting entirely of zero-divisors, Comm. Algebra28 (2000) 1061-1073. · Zbl 0970.13002
[12] Beck, I., Coloring of commutative rings, J. Algebra116 (1988) 208-226. · Zbl 0654.13001
[13] Bell, H. E., Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc.2 (1970) 363-368. · Zbl 0191.02902
[14] Cohn, P. M., reversible rings, Bull. London Math. Soc.31 (1999) 641-648. · Zbl 1021.16019
[15] Feller, E. H., Properties of primary noncommutative rings, Trans. Amer. Math. Soc.89 (1958) 79-91. · Zbl 0095.25703
[16] Fields, D. E., Zero divisors and nilpotents in power series rings, Proc. Amer. Math. Soc.27 (1971) 427-433. · Zbl 0219.13023
[17] Gilmer, R., Grams, A. and Parker, T., Zero divisors in power series rings, J. Reine Angew. Math.1975 (278/279) (1975) 145-164. · Zbl 0309.13009
[18] Habeb, J. M., A note on zero commutative and duo rings, Math. J. Okayama Univ.32 (1990) 73-76. · Zbl 0758.16007
[19] Habibi, M., Moussavi, A. and Alhevaz, A., The McCoy condition on Ore extensions, Comm. Algebra41 (2013) 124-141. · Zbl 1269.16019
[20] Hashemi, E., Abdi, M. and Alhevaz, A., On the diameter of the compressed zero-divisor graph, Comm. Algebra45(11) (2017) 4855-4864. · Zbl 1388.13020
[21] Hashemi, E., Amirjan, R. and Alhevaz, A., On zero-divisor graphs of skew polynomial rings over non-commutative rings, J. Algebra Appl.16(3) (2017), Article ID:1750056, 14 pp. · Zbl 1373.16048
[22] Hashemi, E. and Moussavi, A., Polynomial extensions of quasi-Baer rings, Acta Math. Hungar.107 (2005) 207-224. · Zbl 1081.16032
[23] Henriksen, M. and Jerison, M., The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc.115 (1965) 110-130. · Zbl 0147.29105
[24] Hinkle, G. and Huckaba, J. A., The generalized Kronecker function ring and the ring \(R(X)\), J. Reine Angew. Math.292 (1977) 25-36. · Zbl 0348.13011
[25] Hong, C. Y., Kim, N. K., Lee, Y. and Ryu, S. J., Rings with property \((A)\) and their extensions, J. Algebra315 (2007) 612-628. · Zbl 1156.16001
[26] Huckaba, J. A., Commutative Rings with Zero-Divisors (Marcel Dekker, New York, 1988). · Zbl 0637.13001
[27] Huckaba, J. A. and Keller, J. M., Annihilation of ideals in commutative rings, Pacific J. Math.83 (1979) 375-379. · Zbl 0388.13001
[28] Jordan, D. A., Bijective extensions of injective ring endomorphisms, J. London Math. Soc.25 (1982) 435-448. · Zbl 0486.16002
[29] Kaplansky, I., Commutative Rings, rev. edn. (University of Chicago Press, Chicago, 1974). · Zbl 0203.34601
[30] Kim, N. K. and Lee, Y., Extensions of reversible rings, J. Pure Appl. Algebra210 (2007) 543-550.
[31] Krempa, J., Some examples of reduced rings, Algebra Colloq.3 (1996) 289-300. · Zbl 0859.16019
[32] Kuzmina, A. S. and Maltsev, Yu. N., On finite rings in which zero-divisor graphs satisfy the Diracs condition, Lobachevskii J. Math.36 (2015) 375-383. · Zbl 1344.16021
[33] Lam, T. Y., A first Course in Noncommutative Rings (Springer, New York, 2001). · Zbl 0980.16001
[34] Lucas, T. G., The diameter of a zero divisor graph, J. Algebra301 (2006) 174-193. · Zbl 1109.13006
[35] Maimani, H. R., Pournaki, M. R. and Yassemi, S., Zero-divisor graph with respect to an ideal, Comm. Algebra34 (2006) 923-929. · Zbl 1092.13004
[36] Mason, G., Reflexive ideals, Comm. Algebra9(17) (1981) 1709-1724. · Zbl 0468.16024
[37] de Narbonne, L. Motais, Anneaux semi-commutatifs et unis riels anneaux dont les id aux principaux sont idempotents, in Proc. 106’th National Congress of Learned Societies, Perpignan, , 1981 (Bib. Nat., Paris, 1982), pp. 71-73.
[38] Mulay, S. B., Cycles and symmetries of zero-divisor, Comm. Algebra30 (2002) 3533-3558. · Zbl 1087.13500
[39] Nasr-Isfahani, A. R. and Moussavi, A., Skew Laurent polynomial extensions of baer and p.p.-rings, Bull. Korean Math. Soc.46 (2009) 1041-1050. · Zbl 1188.16023
[40] Quentel, Y., Sur la compacité du spectre minimal d’un anneau, Bull. Soc. Math. France99 (1971) 265-272. · Zbl 0215.36803
[41] Redmond, S. P., The zero-divisor graph of a non-commutative ring, Int. J. Commut. Rings1 (2002) 203-211. · Zbl 1195.16038
[42] Shin, G., Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc.184 (1973) 43-60. · Zbl 0283.16021
[43] Spiroff, S. and Wickham, C., A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra39 (2011) 2338-2348. · Zbl 1225.13007
[44] Tuganbaev, A. A., Semidistributive Modules and Rings, , Vol. 449 (Kluwer Academic Publishers, 1998). · Zbl 0909.16001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.