Geodesic
Description of finite nonnilpotent rings with planar zero-divisor graphs
Diskretnaya Matematika, Tome 21 (2009) no. 4, pp. 60-75.

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The zero-divisor graph of an associative ring $R$ is a graph whose vertices are all nonzero (one-sided and two-sided) zero divisors of $R$, two distinct vertices $x,y$ are connected by an edge if and only if $xy=0$ or $yx=0$. In this paper, all finite nonnilpotent rings with planar zero-divisor graphs are completely described. In the previous paper by Kuzmina and Maltsev, the finite nilpotent rings with planar zero-divisor graphs were studied. Thus, this paper completes the description of finite rings with planar zero-divisor graphs. 
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A. S. Kuzmina. Description of finite nonnilpotent rings with planar zero-divisor graphs. Diskretnaya Matematika, Tome 21 (2009) no. 4, pp. 60-75. http://geodesic.mathdoc.fr/item/DM_2009_21_4_a4/

[1] Beck I., “Coloring of commutative rings”, J. Algebra, 116:1 (1988), 208–226 | DOI | MR | Zbl

[2] Anderson D. F., Livingston P. S., “The zero-divisor graph of a commutative ring”, J. Algebra, 217:2 (1999), 434–447 | DOI | MR | Zbl

[3] Akbari S., Mohammadian A., “On zero-divisor graphs of finite rings”, J. Algebra, 314:1 (2007), 68–184 | DOI | MR

[4] Akbari S., Maimani H. R., Yassemi S., “When zero-divisor graph is planar or a complete $r$-partite graph”, J. Algebra, 270:1 (2003), 169–180 | DOI | MR | Zbl

[5] Belshoff R., Chapman J., “Planar zero-divisor graphs”, J. Algebra, 316:3 (2007), 471–480 | DOI | MR | Zbl

[6] Kuzmina A. S., Maltsev Yu. N., “Nilpotent finite rings with planar zero-divisor graphs”, Asian-European J. Math., 1:4 (2008), 565–574 | DOI | MR

[7] Kharari F., Teoriya grafov, Mir, Moskva, 1973 | MR

[8] Dzhekobson N., Stroenie kolets, IL, Moskva, 1961

[9] Elizarov V. P., Konechnye koltsa, Gelios ARV, Moskva, 2006

[10] Ratinov V. A., Polusovershennye koltsa so spetsialnymi tipami prisoedinennykh grupp, Diss. kand. fiz.-mat. nauk, MGPI, Moskva, 1980

[11] Kuzmina A. S., “O stroenii kolets s planarnymi grafami delitelei nulya”, Izv. Altaiskogo gos. un-ta, 2009, no. 1, 17–25