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Extending abelian groups to rings

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

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
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Abstract

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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
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

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