Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T23:30:24.147Z Has data issue: false hasContentIssue false
  • English
  • Français

THE PROBABILITY OF ZERO MULTIPLICATION IN FINITE RINGS

Part of: Conditions on elements Radicals and radical properties of rings Chain conditions, growth conditions, and other forms of finiteness

Published online by Cambridge University Press:  24 January 2022

DAVID DOLŽAN Open the ORCID record for DAVID DOLŽAN [Opens in a new window]
DAVID DOLŽAN*
Affiliation:
Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, SI-1000 Ljubljana, Slovenia
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

finite ringzero divisorprobability

MSC classification

Primary: 16P10: Finite rings and finite-dimensional algebras
Secondary: 16N40: Nil and nilpotent radicals, sets, ideals, rings 16U99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 1 , August 2022 , pp. 83 - 88
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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), 462–479.Google Scholar
Barry, F., MacHale, D. and Ni She, A., ‘Some supersolvability conditions for finite groups’, Math. Proc. R. Ir. Acad. 106A(2) (2006), 163–177.CrossRefGoogle Scholar
Buckley, S. M. and MacHale, D., ‘Commuting probability for subrings and quotient rings’, J. Algebra Comb. Discrete Struct. Appl. 4(2) (2016), 189–196.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), 372–382.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), 509–528.Google Scholar
Gustafson, W. H., ‘What is the probability that two group elements commute?’, Amer. Math. Monthly 80(9) (1973), 1031–1034.Google Scholar
Kobayashi, Y. and Koh, K., ‘A classification of finite rings by zero divisors’, J. Pure Appl. Algebra 40(2) (1986), 135–147.CrossRefGoogle Scholar
Koh, K., ‘On properties of rings with a finite number of zero divisors’, Math. Ann. 171 (1967), 79–80.Google Scholar
Lescot, P., ‘Isoclinism classes and commutativity degrees of finite groups’, J. Algebra 177 (1995), 847–869.Google Scholar
MacHale, D., ‘Commutativity in finite rings’, Amer. Math. Monthly 83 (1976), 30–32.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), 237–247.Google Scholar