Finite rings in which commutativity is transitive |
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Authors: | David Dol?an Igor Klep Primo? Moravec |
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Institution: | 1.Fakulteta za matematiko in fiziko,Univerza v Ljubljani,Ljubljana,Slovenia;2.Fakulteta za naravoslovje in matematiko,Univerza v Mariboru,Maribor,Slovenia |
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Abstract: | A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings. |
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