Document Type : Research Paper

**Authors**

Department of Mathematics, Shahrood University of Technology, Shahrood, Iran

**Abstract**

Let $R$ be an associative ring with identity and $Z^{\ast}(R)$ be its set of non-zero zero-divisors. Zero-divisor graphs of rings are well represented in the literature of commutative and non-commutative rings. The directed zero-divisor graph of $R$, denoted by $\Gamma{(R)}$, is the directed graph whose vertices are the set of non-zero zero-divisors of $R$ and for distinct non-zero zero-divisors $x,y$, $x\rightarrow y$ is an directed edge if and only if $xy=0$. In this paper, we connect some graph-theoretic concepts with algebraic notions, and investigate the interplay between the ring-theoretical properties of a skew power series ring $R[[x;\alpha]]$ and the graph-theoretical properties of its directed zero-divisor graph $\Gamma(R[[x;\alpha]])$. In doing so, we give a characterization of the possible diameters of $\Gamma(R[[x;\alpha]])$ in terms of the diameter of $\Gamma(R)$, when the base ring $R$ is reversible and right Noetherian with an $\alpha$-condition, namely $\alpha$-compatible property. We also provide many examples for showing the necessity of our assumptions.

**Keywords**

**Main Subjects**

[1] C. Ab dioglu, Zero-divisor graph of matrix rings and Hurwitz rings, Turkish J. Math. , 40 (2016) 201{209.

[2] S. Akbari and A. Mohammadian, The zero-divisor of a commutative ring, J. Algebra , 274 (2004) 847{855.

[3] A. Alhevaz and D. Kiani, On zero divisors in skew inverse Laurent series over non-commutative rings, Comm. Algebra, 42 (2014) 469{487.

[4] A. Alhevaz and D. Kiani, McCoy prop erty of skew Laurent p olynomials and p ower series rings,J. Algebra Appl., 13 (2014) pp. 23.

[5] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra , 217 (1999) 434{447.

[6] D. F. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra , 159 (1993) 500{514.

[7] 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.

[8] S. Annin, Asso ciated prime over skew p olynomials rings, Comm. Algebra , 30 no. 5 (2002) 2511{2528.

[9] M. Axtel, J. Coykendall and J. Stickles, Zero-divisor graph of p olynomials and p ower series over commutative rings, J. Algebra , 33 (2005) 2043{2050.

[10] I. Beck, Coloring of commutative rings, J. Algebra , 116 (1988) 208{226.

[11] D. Bennis, J. Mikram and F. Taraza, On the extended zero divisor graph of commutative rings, Turkish J. Math. , 40 (2016) 376{388.

[12] P. M. Cohn, Reversible rings, Bul l. London Math. Soc. , 31 (1999) 641{648.

[13] D. E. Fields, Zero divisors and nilp otent elements in p ower series rings, Proc. Amer. Math. Soc. , 27 (1971) 427{433.

[14] E. Hashemi, Compatible ideals and radicals of Ore extensions, New York J. Math. , 12 (2006) 349{356.

[15] E. Hashemi, A. Alhevaz and E. Yo onesian, On zero divisor graph of unique pro duct monoid rings over No etherian reversible ring, Categories and General Algebraic Structures with Applications , 4 (2015) 95{114.

[16] E. Hashemi, and R. Amirjan, Zero-divisor graphs of Ore extensions over reversible rings, Canad. Math. Bul l. , 59 (2016) 794{805

[17] E. Hashemi, R. Amirjan and A. Alhevaz, On zero-divisor graphs of skew p olynomial rings over non-commutative rings, J. Algebra Appl. , 16 (2017) pp. 14.

[18] E. Hashemi, A. As. Esta ji and M. Ziemb owski, Answers to some questions concerning rings with prop erty (A). Proc. Edinb. Math. Soc. , 60 (2017) 651{664.

[19] E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Bear rings, Acta Math. Hunger , 107 (2005) 207{224.

[20] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc., 115 (1965) 110{130.

[21] C. Y. Hong, N. K. Kim, Y. Lee and S. J. Ryu, Rings with prop erty (A) and their extensions. J. Algebra , 315 no. 2

(2007) 612{628.

[22] J. A. Huckaba, and J. M. Keller, Annihilation of ideals in commutative rings, Pacic J. Math. , 83 no. 2 (1979) 375{379.

[23] N. K. Kim, and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra , 210 no 1-3 (2007) 543{550.

[24] J. Krempa, and D. Niewieczerzal, Rings in which annihilators are ideals and their application to semigroup rings, Bul l. Acad. Polon. Sci. Ser. Math. Astronom. Phys. , 25 (1977) 851{856.

[25] J. Krempa, Examples of reduced ring, Algebra Col loq. , 3 no. 4 (1996) 289{300.

[26] T. Lucas, The diameter of a zero divisor graph, J. Algebra , 301 (2006) 174{193.

[27] Y. Quentel, Sur la compacite' du sp ectre minimal d'un anneau, Bul l. Soc. Math. France , 99 (1971) 265{272.

[28] S. P. Redmond, The zero-divisor graph of a non-commmutative ring, Int. J. Commut. Rings , 1 (2002) 203{211.

[29] S. P. Redmond, Structure in the zero-divisor graph of a non-commutative ring, Houston J. Math. 30 no. 2 (2004)

345{355.

[2] S. Akbari and A. Mohammadian, The zero-divisor of a commutative ring, J. Algebra , 274 (2004) 847{855.

[3] A. Alhevaz and D. Kiani, On zero divisors in skew inverse Laurent series over non-commutative rings, Comm. Algebra, 42 (2014) 469{487.

[4] A. Alhevaz and D. Kiani, McCoy prop erty of skew Laurent p olynomials and p ower series rings,J. Algebra Appl., 13 (2014) pp. 23.

[5] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra , 217 (1999) 434{447.

[6] D. F. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra , 159 (1993) 500{514.

[7] 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.

[8] S. Annin, Asso ciated prime over skew p olynomials rings, Comm. Algebra , 30 no. 5 (2002) 2511{2528.

[9] M. Axtel, J. Coykendall and J. Stickles, Zero-divisor graph of p olynomials and p ower series over commutative rings, J. Algebra , 33 (2005) 2043{2050.

[10] I. Beck, Coloring of commutative rings, J. Algebra , 116 (1988) 208{226.

[11] D. Bennis, J. Mikram and F. Taraza, On the extended zero divisor graph of commutative rings, Turkish J. Math. , 40 (2016) 376{388.

[12] P. M. Cohn, Reversible rings, Bul l. London Math. Soc. , 31 (1999) 641{648.

[13] D. E. Fields, Zero divisors and nilp otent elements in p ower series rings, Proc. Amer. Math. Soc. , 27 (1971) 427{433.

[14] E. Hashemi, Compatible ideals and radicals of Ore extensions, New York J. Math. , 12 (2006) 349{356.

[15] E. Hashemi, A. Alhevaz and E. Yo onesian, On zero divisor graph of unique pro duct monoid rings over No etherian reversible ring, Categories and General Algebraic Structures with Applications , 4 (2015) 95{114.

[16] E. Hashemi, and R. Amirjan, Zero-divisor graphs of Ore extensions over reversible rings, Canad. Math. Bul l. , 59 (2016) 794{805

[17] E. Hashemi, R. Amirjan and A. Alhevaz, On zero-divisor graphs of skew p olynomial rings over non-commutative rings, J. Algebra Appl. , 16 (2017) pp. 14.

[18] E. Hashemi, A. As. Esta ji and M. Ziemb owski, Answers to some questions concerning rings with prop erty (A). Proc. Edinb. Math. Soc. , 60 (2017) 651{664.

[19] E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Bear rings, Acta Math. Hunger , 107 (2005) 207{224.

[20] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc., 115 (1965) 110{130.

[21] C. Y. Hong, N. K. Kim, Y. Lee and S. J. Ryu, Rings with prop erty (A) and their extensions. J. Algebra , 315 no. 2

(2007) 612{628.

[22] J. A. Huckaba, and J. M. Keller, Annihilation of ideals in commutative rings, Pacic J. Math. , 83 no. 2 (1979) 375{379.

[23] N. K. Kim, and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra , 210 no 1-3 (2007) 543{550.

[24] J. Krempa, and D. Niewieczerzal, Rings in which annihilators are ideals and their application to semigroup rings, Bul l. Acad. Polon. Sci. Ser. Math. Astronom. Phys. , 25 (1977) 851{856.

[25] J. Krempa, Examples of reduced ring, Algebra Col loq. , 3 no. 4 (1996) 289{300.

[26] T. Lucas, The diameter of a zero divisor graph, J. Algebra , 301 (2006) 174{193.

[27] Y. Quentel, Sur la compacite' du sp ectre minimal d'un anneau, Bul l. Soc. Math. France , 99 (1971) 265{272.

[28] S. P. Redmond, The zero-divisor graph of a non-commmutative ring, Int. J. Commut. Rings , 1 (2002) 203{211.

[29] S. P. Redmond, Structure in the zero-divisor graph of a non-commutative ring, Houston J. Math. 30 no. 2 (2004)

345{355.

[30] Sh. Yang, X. Song, and Zh. Liu, Power-serieswise McCoy Rings. Algebra Col loq. , 18 no. 2 (2011) 301{310.

December 2018

Pages 43-57

**Receive Date:**10 January 2018**Revise Date:**26 September 2018**Accept Date:**17 October 2018**Published Online:**01 December 2018