Zero-divisor graphs of amalgamations
DOI:
https://doi.org/10.7146/math.scand.a-105307Abstract
Let $f\colon A\rightarrow B$ be a homomorphism of commutative rings and let $J$ be an ideal of $B$. The amalgamation of $A$ with $B$ along $J$ with respect to $f$ is the subring of $A\times B$ given by \[ A\bowtie ^{f}J:=\{(a,f(a)+j) \mid a\in A, j\in J\}. \] This paper investigates the zero-divisor graph of amalgamations. Our aim is to characterize when the graph is complete and compute its diameter and girth for various contexts of amalgamations. The new results recover well-known results on duplications, and yield new and original examples issued from amalgamations.
References
Anderson, D. D., Commutative rings, in “Multiplicative ideal theory in commutative algebra”, Springer, New York, 2006, pp. 1--20. https://doi.org/10.1007/978-0-387-36717-0_1
Anderson, D. D. and Naseer, M., Beck's coloring of a commutative ring, J. Algebra 159 (1993), no. 2, 500–514. https://doi.org/10.1006/jabr.1993.1171
Anderson, D. F. and Livingston, P. S., The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434–447. https://doi.org/10.1006/jabr.1998.7840
Anderson, D. F. and Mulay, S. B., On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra 210 (2007), no. 2, 543–550. https://doi.org/10.1016/j.jpaa.2006.10.007
Axtell, M., Coykendall, J., and Stickles, J., Zero-divisor graphs of polynomials and power series over commutative rings, Comm. Algebra 33 (2005), no. 6, 2043–2050. https://doi.org/10.1081/AGB-200063357
Axtell, M. and Stickles, J., Zero-divisor graphs of idealizations, J. Pure Appl. Algebra 204 (2006), no. 2, 235–243. https://doi.org/10.1016/j.jpaa.2005.04.004
Beck, I., Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226. https://doi.org/10.1016/0021-8693(88)90202-5
D'Anna, M., A construction of Gorenstein rings, J. Algebra 306 (2006), no. 2, 507–519. https://doi.org/10.1016/j.jalgebra.2005.12.023
D'Anna, M., Finocchiaro, C. A., and Fontana, M., Amalgamated algebras along an ideal, in “Commutative algebra and its applications”, Walter de Gruyter, Berlin, 2009, pp. 155--172.
D'Anna, M., Finocchiaro, C. A., and Fontana, M., Properties of chains of prime ideals in an amalgamated algebra along an ideal, J. Pure Appl. Algebra 214 (2010), no. 9, 1633–1641. https://doi.org/10.1016/j.jpaa.2009.12.008
D'Anna, M. and Fontana, M., The amalgamated duplication of a ring along a multiplicative-canonical ideal, Ark. Mat. 45 (2007), no. 2, 241–252. https://doi.org/10.1007/s11512-006-0038-1
D'Anna, M. and Fontana, M., An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl. 6 (2007), no. 3, 443–459. https://doi.org/10.1142/S0219498807002326
DeMeyer, F. and Schneider, K., Automorphisms and zero divisor graphs of commutative rings, in “Commutative rings”, Nova Sci. Publ., Hauppauge, NY, 2002, pp. 25--37.
Dorroh, J. L., Concerning adjunctions to algebras, Bull. Amer. Math. Soc. 38 (1932), no. 2, 85–88. https://doi.org/10.1090/S0002-9904-1932-05333-2
Lucas, T. G., The diameter of a zero divisor graph, J. Algebra 301 (2006), no. 1, 174–193. https://doi.org/10.1016/j.jalgebra.2006.01.019
Maimani, H. R. and Yassemi, S., Zero-divisor graphs of amalgamated duplication of a ring along an ideal, J. Pure Appl. Algebra 212 (2008), no. 1, 168–174. https://doi.org/10.1016/j.jpaa.2007.05.015
Mulay, S. B., Cycles and symmetries of zero-divisors, Comm. Algebra 30 (2002), no. 7, 3533–3558. https://doi.org/10.1081/AGB-120004502
Nagata, M., Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers, John Wiley & Sons, New York-London, 1962.
