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THE PROBABILITY OF ZERO MULTIPLICATION IN FINITE RINGS

Published online by Cambridge University Press:  24 January 2022

DAVID DOLŽAN*
Affiliation:
Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, SI-1000 Ljubljana, Slovenia
*

Abstract

Let R be a finite ring and let ${\mathrm {zp}}(R)$ denote the nullity degree of R, that is, the probability that the multiplication of two randomly chosen elements of R is zero. We establish the nullity degree of a semisimple ring and find upper and lower bounds for the nullity degree in the general case.

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The author acknowledges financial support from the Slovenian Research Agency (research core funding no. P1-0222).

References

Akbari, S. and Mohammadian, A., ‘Zero-divisor graphs of non-commutative rings’, J. Algebra 296(2) (2006), 462479.Google Scholar
Barry, F., MacHale, D. and Ni She, A., ‘Some supersolvability conditions for finite groups’, Math. Proc. R. Ir. Acad. 106A(2) (2006), 163177.CrossRefGoogle Scholar
Buckley, S. M. and MacHale, D., ‘Commuting probability for subrings and quotient rings’, J. Algebra Comb. Discrete Struct. Appl. 4(2) (2016), 189196.Google Scholar
Buckley, S. M., MacHale, D. and Ni She, A., ‘Finite rings with many commuting pairs of elements’, Preprint, 2014. Available online at https://archive.maths.nuim.ie/staff/sbuckley/Papers/bms.pdf.Google Scholar
Dutta, J., Basnet, D. K. and Nath, R. K., ‘On commuting probability of finite rings’, Indag. Math. (N.S.) 28(2) (2017), 372382.CrossRefGoogle Scholar
Esmkhani, M. A. and Jafarian Amiri, S. M., ‘The probability that the multiplication of two ring elements is zero’, J. Algebra Appl. 17(3) (2018), 1850054.CrossRefGoogle Scholar
Esmkhani, M. A. and Jafarian Amiri, S. M., ‘ Characterization of rings with nullity degree at least  $1/4$ ’, J. Algebra Appl. 18(4) (2019), 1950076.CrossRefGoogle Scholar
Guralnick, R. M. and Robinson, G. R., ‘On the commuting probability in finite groups’, J. Algebra 300 (2006), 509528.Google Scholar
Gustafson, W. H., ‘What is the probability that two group elements commute?’, Amer. Math. Monthly 80(9) (1973), 10311034.Google Scholar
Kobayashi, Y. and Koh, K., ‘A classification of finite rings by zero divisors’, J. Pure Appl. Algebra 40(2) (1986), 135147.CrossRefGoogle Scholar
Koh, K., ‘On properties of rings with a finite number of zero divisors’, Math. Ann. 171 (1967), 7980.Google Scholar
Lescot, P., ‘Isoclinism classes and commutativity degrees of finite groups’, J. Algebra 177 (1995), 847869.Google Scholar
MacHale, D., ‘Commutativity in finite rings’, Amer. Math. Monthly 83 (1976), 3032.CrossRefGoogle Scholar
McDonald, B. R., Finite Rings with Identity, Pure and Applied Mathematics, 28 (Marcel Dekker, New York, 1974).Google Scholar
Morrison, K. E., ‘Integer sequences and matrices over finite fields’, J. Integer Seq. 9(2) (2006), 06.2.1.Google Scholar
Rusin, D., ‘What is the probability that two elements of a finite group commute?’, Pacific J. Math. 82(1) (1979), 237247.Google Scholar