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MINIMAL SETS IN FINITE RINGS

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MINIMAL SETS IN FINITE RINGS LEANNE CONAWAY AND KEITH A KEARNES Abstract We describe how to calculate the h i minimal sets in any finite ring 1 Introduction In the mid 1980 s David Hobby and Ralph McKenzie introduced a general localization theory for algebras called tame congruence theory The theory has proven to be an effective tool for analyzing the structure of finite algebras and locally finite varieties A preliminary version of the theory appears in 13 the handbook of the theory is 4 and later expositions of the theory can be found in 1 2 5 8 9 10 One of the strengths of tame congruence theory is that its concepts are languageindependent But this feature raises a question about how the theory might make connections with classical algebra Since the theory studies finite algebras via the structure and distribution of their minimal sets this question might be phrased more specifically as How do you calculate the minimal sets in a finite group ring module semigroup etc One of the problems posed by Hobby and McKenzie is Problem 15 of 4 Investigate h0 i minimal sets for abelian minimal congruences of finite groups Do the same for finite rings In this paper we solve the second half of Problem 15 by describing all minimal sets in finite rings1 up to polynomial isomorphism As part of the solution we find it necessary to describe the minimal sets in finite modules and bimodules The main results are Theorems 2 9 3 5 4 1 and Corollary 4 2 The part of Problem 15 concerning finite groups seems harder than the part about rings and it is still open A partial result in this direction was proved in 1996 by K Kearnes E W Kiss and C Szabo if G is a finite group then any p Sylow subgroup of G is a neighborhood See the next section for this definition See 8 for a proof of this statement about groups This result solves the part of Problem 15 that concerns nilpotent groups 1991 Mathematics Subject Classification Primary 08B05 Secondary 16P10 Key words and phrases finite ring minimal

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