Shafiq Ur REHMAN, Muhammad Naveed SHAHERYAR

On generalized probability in finite commutative rings

Let $R$ be a finite commutative ring with unity and $x\in R$. We study the probability that the product of two randomly chosen elements (with replacement) of $R$ equals $x$. We denote this probability by $Prob_x (R)$. We determine some bounds for this probability and also obtain some characterizations of finite commutative rings based on this probability. Moreover, we determine the explicit computing formulas for $Prob_x (R)$ when $R=\mathbb{Z}_m\times \mathbb{Z}_n$.

Keywords:

Probability, finite commutative ring, Euler-phi function, simple graph, zero-divisor graph complete graph, local ring,

PDF

___

M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., 1969.
D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, Journal of Algebra, 217(2) (1999), 434-447.
D. F. Anderson, A. Frazier, A. Lauve and P. S. Livingston, The zero-divisor graph of a commutative ring II, In: Ideal theoretic methods in commutative algebra (Columbia (MO); 1999), Lecture Notes in Pure and Applied Mathematics, vol. 220, Dekker, New York, 2001, pp. 61-72.
S. M. Buckley and D. Machale, Commuting probability for subrings and quotient rings, J. Algebra Comb. Discrete Struct. Appl., 4(2) (2017), 189-196.
D.S. Dummit and R. M. Foote, Abstract Algebra, third edition, John Wiley and Sons, Inc., Hoboken, NJ, 2004.
P. Erdos and P. Turan, On some problems of a statistical group theory IV, Acta Math. Acad. Sci. Hungar., 19 (1968), 413-435.
M. A. Esmkhani and S. M. Jafarian Amiri, The probability that the multiplication of two ring elements is zero, J. Algebra Appl., 17(3) (2018), 9 pp.
W. H. Gustafson, What is the probability that two group elements commute?, Amer. Math. Monthly, 80 (1973), 1031-1034.
Sanhan M.S. Khasraw, What is the probability that two elements of a finite ring have product zero?, Mal. J. Fund. Appl. Sci., 16(04) (2020), 497-499.
D. MacHale, How commutative can a non-commutative group be?, Math. Gaz., 58 (1974), 199-202.
D. Machale, Commutativity in finite rings, Amer. Math. Monthly, 83(1) (1975), 30-32.
S.P. Redmond, The zero-divisor graph of a non-commutative ring, In: Commutative rings, Nova Sci. Publ., Hauppauge, NY, 2002, 39-47.
S.P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Comm. Algebra, 31(9) (2003), 4425-4443.
S.P. Redmond, Structure in the zero-divisor graph of a noncommutative ring, Houston J. Math, 30(2) (2004), 345-355.
S.P. Redmond, On zero-divisor graphs of small finite commutative rings, Discrete Math., 307 (2007), 1155-1166.
S. U. Rehman, A. Q. Baig and K. Haider, A probabilistic approach toward finite commutative ring, Southeast Asian Bull. Math., 43 (2019), 413-418.
D. J. Rusin, What is the probability that two elements of a finite group commute?, Pacific J. Math., 82 (1979), 237-247.