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For any finite abelian group (R,+), we define a binary operation or “multiplication” on R and give necessary and sufficient conditions on this multiplication for R to extend to a ring. Then we show when two rings made on the same group are isomorphic. In particular, it is shown that there are n+1 rings of order pn with characteristic pn, where p is a prime number. Also, all finite rings of order p6 are described by generators and relations. Finally, we give an algorithm for the computation of all finite rings based on their additive group.