×

Zero-divisor graphs of idealizations. (English) Zbl 1104.13003

The authors of the reviewed paper (with J. Coykendall) examined in [Commun. Algebra 33, No. 6, 2043–2050 (2005; Zbl 1088.13006)] the preservation, or lack thereof, of the diameter and girth of the graph of a commutative ring under extensions to polynomial and power series rings.
In the paper under review they look at the preservation of the diameter and girth under idealizations of commutative rings. Specifically, in section 2 they completely characterize the girth of the zero-divisor graph in the idealization. In section 3 they completely characterize when the zero-divisor graph of an idealization will be complete and provide some conditions when the zero-divisor graph of the idealization will have diameter \(2\). They also present some questions that remain.

MSC:

13A99 General commutative ring theory
13B99 Commutative ring extensions and related topics
13F20 Polynomial rings and ideals; rings of integer-valued polynomials

Citations:

Zbl 1088.13006
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anderson, D. F.; Frazier, A.; Lauve, A.; Livingston, P., The Zero Divisor Graph of a Commutative Ring, II, (Lecture Notes in Pure and Applied Mathematics, vol. 220 (2001), Dekker: Dekker New York), 61-72 · Zbl 1035.13004
[2] Anderson, D. F.; Livingston, P. S., The zero-divisor graph of a commutative ring, J. Algebra, 217, 434-447 (1999) · Zbl 0941.05062
[3] Anderson, D. D.; Naseer, M., Beck’s coloring of a commutative ring, J. Algebra, 159, 500-514 (1993) · Zbl 0798.05067
[4] M. Axtell, J. Coykendall, J. Stickles, Zero-divisor graphs of polynomial and power series over commutative rings, Comm. Algebra, 33 (6) (2005), to appear.; M. Axtell, J. Coykendall, J. Stickles, Zero-divisor graphs of polynomial and power series over commutative rings, Comm. Algebra, 33 (6) (2005), to appear. · Zbl 1088.13006
[5] Beck, I., Coloring of commutative rings, J. Algebra, 116, 208-226 (1988) · Zbl 0654.13001
[6] DeMeyer, F.; Schneider, K., Automorphisms and zero-divisor graphs of commutative rings, Int. J. Commut. Rings, 1, 3, 93-106 (2002)
[7] Fields, D. E., Zero divisors and nilpotent elements in power series, Proc. Amer. Math. Soc., 27, 3, 427-433 (1971) · Zbl 0219.13023
[8] Huckaba, J., Commutative rings with zero divisors, (Monographs Pure Applied Mathematics (1988), Marcel Dekker: Marcel Dekker Basel, New York) · Zbl 0637.13001
[9] Mulay, S. B., Cycles and symmetries of zero-divisors, Comm. Algebra, 30, 7, 3533-3558 (2002) · Zbl 1087.13500
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.