J. Aust. Math. Soc.  82 (2007), 297-313
Extending abelian groups to rings

Lynn M. Batten
  School of Computing and Mathematics
  Deakin University
  221 Burwood Highway
  Burwood Vic 3125
Robert S. Coulter
  Department of Mathematical Sciences
  520 Ewing Hall
  University of Delaware
  Newark, Delaware 19716
Marie Henderson
  307/60 Willis Street
  Te Aro (Wellington), 6001
  New Zealand

For any abelian group G and any function $f: G \rightarrow G$ 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.
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