Abstract
For any finite abelian group(R,+), we define a binary operation or âmultiplicationâ 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.
Highlights
Roughout the paper, all rings are associative
Instead of modules over a commutative ring, we focus our attention to the Z-modules
Let RR be a ring of order ppn1n1 ppn2n2 ⯠ppnknkkk, where the ppii are distinct primes and the nnii are positive integers. en RR is expressible, in a where |RRii| = ppniinii unique manner, ii iiiii i iii
Summary
E problem of determining and classifying up to isomorphism nite rings has received considerable attention, both old and new, see [1,2,3,4]. In [9], the authors introduced a multiplication (similar to eorem 3) on a nitely generated SS-module to extend to an SS-algebra, where SS is a commutative ring. A number of necessary and sufficient conditions for any SS-module to extend to an SS-algebra, where SS is a commutative ring, is given by Behboodi et al [9]. We prove that there are nn n n rings (up to isomorphism) of order ppnn whose abelian group is cyclic. All rings of order pp whose abelian group is Zpp â Zpp are determined.
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