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Extending abelian groups to rings

Published online by Cambridge University Press:  09 April 2009

Lynn M. Batten
Affiliation:
School of Computing and MathematicsDeakin University221 Burwood HighwayBurwood Vic [email protected]
Robert S. Coulter
Affiliation:
Department of Mathematical Sciences520 Ewing HallUniversity of DelawareNewark, Delaware [email protected]
Marie Henderson
Affiliation:
307/60 Willis StreetTe Aro (Wellington), 6001New [email protected]
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Abstract

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For any abelian group G and any function f: GG 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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2007

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