Rings over which polynomial rings are Armendariz and reversible
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A ring R is called reversibly Armendariz if bjai = 0for all i; j whenever f(x)g(x) Σ = 0 for two polynomials f(x) = m i=0 aixi; g(x) =Σn j=0 bjxj over R. It is proved that a ring R is reversibly Armendariz if and only if its polynomial ring is re-versibly Armendariz if and only if its Laurent polynomial ring is re-versibly Armendariz. Relations between reversibly Armendariz rings and related ring properties are examined in this note, observing the structures of many examples concerned. Various kinds of reversibly Armendariz rings are provided in the process. Especially it is shown to be possible to construct reversibly Armendariz rings from given any Armendariz rings.
Abstract
A ring R is called reversibly Armendariz if bjai = 0for all i; j whenever f(x)g(x) Σ = 0 for two polynomials f(x) = m i=0 aixi; g(x) =Σn j=0 bjxj over R. It is proved that a ring R is reversibly Armendariz if and only if its polynomial ring is re-versibly Armendariz if and only if its Laurent polynomial ring is re-versibly Armendariz. Relations between reversibly Armendariz rings and related ring properties are examined in this note, observing the structures of many examples concerned. Various kinds of reversibly Armendariz rings are provided in the process. Especially it is shown to be possible to construct reversibly Armendariz rings from given any Armendariz rings.
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