On weak Armendariz rings
Bull. Korean Math. Soc. 2009 Vol. 46, No. 1, 135-146
Printed January 1, 2009
Young Cheol Jeon, Hong Kee Kim, Yang Lee, and Jung Sook Yoon
Korea Science Academy, Gyeongsang National University, Busan National University, and Busan National University
Abstract : In the present note we study the properties of weak Armendariz rings, and the connections among weak Armendariz rings, Armendariz rings, reduced rings and IFP rings. We prove that a right Ore ring $R$ is weak Armendariz if and only if so is $Q$, where $Q$ is the classical right quotient ring of $R$. With the help of this result we can show that a semiprime right Goldie ring $R$ is weak Armendariz if and only if $R$ is Armendariz if and only if $R$ is reduced if and only if $R$ is IFP if and only if $Q$ is a finite direct product of division rings, obtaining a simpler proof of Lee and Wong's result. In the process we construct a semiprime ring extension that is infinite dimensional, from given any semiprime ring. We next find more examples of weak Armendariz rings.
Keywords : Armendariz ring, weak Armendariz ring, reduced ring, IFP ring, classical quotient ring, semiprime ring, abelian ring, Goldie ring
MSC numbers : 16N60, 16S36, 16U20
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