In a 1992 paper by the authors we encountered the problem of finding finite ringswith identity of which the lattice of the left ideals is not isomorphic to the lattice of the right ideals. Such rings are, of course, far from commutative. We have at least to demand that these rings do not have any anti-automorphism (a bijection phi of R onto R such that phi(a + b) = phi(a) + phi(b) and phi(ab) = phi(b)phi(a) for all a, b a member of R). We call a ring with an anti-automorphism anti-automorphic. This problem leads, in a natural way, to a classification of the noncommutative rings with identity. We restrict ourselves to such rings of a cardinality dividing p(sup 4), where p is prime. In this report all rings are associative and rings denoted by R will be finite and contain an identity. It is easily shown that any finite ring is the direct sum of rings of prime power cardinality. If the ring has an identity, then all the direct summands have an identity. As a consequence we shall restrict ourselves in our classification to rings with identity of prime-power cardinality, that is we are interested in so-called p-rings (with identity); these are rings of cardinality p(sup k), with p prime and k a natural number.
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