J. Aust. Math. Soc.
82 (2007), 297313

Extending abelian groups to rings

Lynn M. Batten
School of Computing and Mathematics
Deakin University
221 Burwood Highway
Burwood Vic 3125
Australia
lmbatten@deakin.edu.au


Robert S. Coulter
Department of Mathematical Sciences
520 Ewing Hall
University of Delaware
Newark, Delaware 19716
USA
coulter@math.udel.edu





Abstract

For any abelian group
G
and any function
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
pgroup 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
pgroup of odd order
p^{2}.

Download the article in PDF format (size 168 Kb)


Australian Mathematical Publishing Association Inc.

©
Australian MS

