Extending abelian groups to rings

For any abelian group G and any function<i> f : G → G</i> we define a commutative binary operation or `multiplication' on <i>G</i> in terms of <i>f</i>. We give necessary and sufficient conditions on <i>f </i>for <i>G </i>to extend to a commutative ring with the new multiplication. In the case where <i>G </i>is an elementary abelian <i>p</i>-group of odd order, we classify those functions which extend <i>G </i>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 <i>p²</i>. <br />