Article in volume

Authors:

S. Visweswaran

Subramanian Visweswaran

Saurashtra Universit

email: s_visweswaran2006@yahoo.co.in

0000-0002-4905-809X

Title:

Some remarks on the dominating sets of the annihilating-ideal graph of a commutative ring

Source:

Discussiones Mathematicae - General Algebra and Applications 44(2) (2024) 383-412

Received: 2023-01-30 , Revised: 2023-07-08 , Accepted: 2023-07-10 , Available online: 2024-08-19 , https://doi.org/10.7151/dmgaa.1458

Abstract:

Primary keywords:

Reduced ring, Maximal N-prime of (0), Minimal prime ideal, Annihilating ideal

Secondary keywords:

Dominating sets, Domination number of a graph, Total domination number of a graph, Annihilating-ideal graph

References:

1. A. Alilou and J. Amjadi, The sum-annihilating essential ideal graph of a commutative ring, Commun. Comb. Optimization 1 (2) (2016) 117-135.
   https://doi.org/22049/CCO.2016.13555
2. M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra (Addison-Wesley, Massachusetts, 1969).
3. R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory (Second Edition) (Universitext Springer, New York, 2012).
4. I. Beck, Coloring of commutative rings, J. Algebra 116 (1) (1988) 208-226.
   https://doi.org/10.1016/0021-8693(88)90202-5
5. M. Behboodi and Z. Rakeei, Annihilating-ideal graph of commutative rings $I$, J.Algebra Appl. 10 (4) (2011) 727-739.
   https://doi.org/10.1142/S0219498811004896
6. M. Behboodi and Z. Rakeei, Annihilating-ideal graph of commutative rings $II$, J. Algebra Appl. 10 (4) (2011) 741-753.
   https://doi.org/10.1142/S0219498811004902
7. D.E. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc. 27 (3) (1971) 427-433.
   https://doi.org/10.1090/S0002-9939-1971-0271100-6
8. R. Gilmer, Multiplicative Ideal Theory (Marcel Dekker, New York, 1972).
9. R. Gilmer and W. Heinzer, The Laskerian property, power series rings and Noetherian spectra, Proc. Amer. Math. Soc. 79 (1) (1980) 13-16.
   https://doi.org/10.1090/S0002-9939-1980-0560575-6
10. W. Heinzer and D. Lantz, The Laskerian property in commutative rings, J. Algebra 72 (1) (1981) 101-114.
    https://doi.org/10.1016/0021-8693(81)90313-6
11. W. Heinzer and J. Ohm, On the Noetherian-like rings of E.G. Evans, Proc. Amer. Math. Soc. 34 (1) (1972) 73-74.
    https://doi.org/10.1090/S0002-9939-1972-0294316-2
12. I. Kaplansky, Commutative Rings (The University of Chicago Press, Chicago, 1974).
13. H. B. Mann, Introduction to Algebraic Number Theory (Ohio State University Press, Columbus, Ohio, 1955).
14. R. Nikandish and H. R. Maimani, Dominating sets of Annihilating-ideal graphs, Electronic Notes in Disc. Math. 45 (2014) 17-22.
    https://doi.org/10.1016/j.endm.2013.11.005
15. R. Nikandish, H.R. Maimani and S. Kiani, Domination number in the annihilating-ideal graphs of commutative rings, Publ. de l'Institut Math. Nouvelle serie {tome 97 111} (2015) 225-231.
    https://doi.org/10.2298/PIM140222001N
16. N. Kh. Tohidi, M.J. Nikmehr and R. Nikandish, On the strongly annihilating-ideal graph of a commutative ring, Discrete Math. Algorithms Appl. 9 (2) (2017) Art ID:1750028 (13 pages).
    https://doi.org/10.1142/S1793830917500288
17. S. Visweswaran and Premkumar T. Lalchandani, The exact zero-divisor graph of a reduced ring, Indian J. Pure Appl. Math. 52 (4) (2021) 1123-1144.
    https://doi.org/10.1007/s13226-021-00086-9