Classification of Finite Rings of Orderp6by Generators and Relations

摘要

For any finite abelian group(R,+), we define a binary operation or “multiplication” onRand give necessary and sufficient conditions on this multiplication forRto extend to a ring. Then we show when two rings made on the same group are isomorphic. In particular, it is shown that there aren+1rings of orderpnwith characteristicpn, wherepis a prime number. Also, all finite rings of orderp6are described by generators and relations. Finally, we give an algorithm for the computation of all finite rings based on their additive group.