Hostname: page-component-7dd5485656-bvgqh Total loading time: 0 Render date: 2025-10-31T06:48:01.335Z Has data issue: false hasContentIssue false
  • English
  • Français

Extending abelian groups to rings

Part of: General commutative ring theory Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  09 April 2009

Lynn M. Batten ,
Robert S. Coulter  and
Marie Henderson
Lynn M. Batten
Affiliation:
School of Computing and MathematicsDeakin University221 Burwood HighwayBurwood Vic 3125Australialmbatten@deakin.edu.au
Robert S. Coulter
Affiliation:
Department of Mathematical Sciences520 Ewing HallUniversity of DelawareNewark, Delaware 19716USAcoulter@math.udel.edu
Marie Henderson
Affiliation:
307/60 Willis StreetTe Aro (Wellington), 6001New Zealandmarie.henderson@ssc.govt.nz
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For any abelian group G and any function f: G → G we define a commutative binary operation or ‘multiplication’ on G in terms of f. We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p–group of odd order, we classify those functions which extend G to a ring and show, under an equivalence relation we call weak isomorphism, that there are precisely six distinct classes of rings constructed using this method with additive group the elementary abelian p–group of odd order p2.

MSC classification

Secondary: 11T30: Structure theory 13A99: None of the above, but in this section

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 3 , June 2007 , pp. 297 - 314
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Carlitz, L., ‘Invariantive theory of equations in a finite field’, Trans. Amer. Math. Soc. 75 (1953), 405–427.CrossRefGoogle Scholar
[2]Carlitz, L., ‘Invariant theory of systems of equations in a finite field’, J. Analyse Math. 3 (1954). 382–413.CrossRefGoogle Scholar
[3]Cavior, S. R., ‘Equivalence classes of functions over a finite field’, Acta Arith. 10(1964), 119–136.CrossRefGoogle Scholar
[4]Cavior, S. R., ‘Equivalence classes of sets of polynomials over a finite field’, J. Reine Angew. Math. 225 (1967), 191–202.Google Scholar
[5]Corbas, B. and Williams, G. D., ‘Rings of order p 5. Part I. Nonlocal rings’, J. Algebra 231 (2000), 677–690.CrossRefGoogle Scholar
[6]Corbas, B., ‘Rings of order p 5. Part II. Local rings’, J. Algebra 231 (2000), 691–704.CrossRefGoogle Scholar
[7]Coulter, R. S. and Matthews, R. W., ‘Planar functions and planes of Lenz-Barlotti class II’, Des. Codes Cryptogr. 10 (1997), 167–184.CrossRefGoogle Scholar
[8]Dembowski, P., Finite geometries (Springer, New York, 1968, reprinted 1997).CrossRefGoogle Scholar
[9]Dembowski, P. and Ostrom, T. G., ‘Planes of order n with collineation groups of order n 2, Math. Z. 103(1968), 239–258.CrossRefGoogle Scholar
[10]Dickson, L. E., ‘General theory of modular invariants’, Trans. Amer. Math. Soc. 10 (1909), 123–158.CrossRefGoogle Scholar
[11]Goodaire, E. G. and Zhou, Y., ‘Alternative rings of small order’, Comm. Algebra 28 (2000), 3335–3349.CrossRefGoogle Scholar
[12]Herstein, I. N., Topics in algebra, 2nd edition (John Wiley & Sons, 1975).Google Scholar
[13]Knuth, D. E., ‘Finite semifields and projective planes’, J. Algebra 2 (1965), 182–217.CrossRefGoogle Scholar
[14]Lidl, R., Mullen, G. L. and Turnwald, G., Dickson polynomials, Pitman Monographs and Surveys in Pure and Appl. Math. 65 (Longman Scientific and Technical, Essex, England, 1993).Google Scholar
[15]Lidl, R. and Niederreiter, H., Finite fields, Encyclopedia Math. Appl., vol. 20 (Addison-Wesley, Reading, 1983, (now distributed by Cambridge University Press)).Google Scholar
[16]Mullen, G. L., ‘Equivalence classes of functions over a finite field’, Acta Arith. 29 (1976), 353–358.CrossRefGoogle Scholar
[17]Mullen, G. L., ‘Equivalence classes of polynomials over finite fields’, Acta Arith. 31 (1976), 113–123.CrossRefGoogle Scholar
[18]Mullen, G. L., ‘Weak equivalence of functions over a finite field’, Acta Arith. 35 (1979), 259–272.CrossRefGoogle Scholar
[19]Williams, G. D., ‘On a class of finite rings of characteristic p 2’, Result. Math. 38 (2000), 377–390.CrossRefGoogle Scholar