# The $w$-FF property in trivial extensions

Document Type: Research Paper

Authors

1 Department of Mathematics Education‎, ‎Incheon National University‎, ‎Incheon 22012‎, ‎Republic of Korea.

2 School of Computer and Information Engineering‎, ‎Hoseo University‎, ‎Asan 31499‎, ‎Republic of Korea.

Abstract

‎Let $D$ be an integral domain with quotient field $K$‎, ‎$E$ be a $K$-vector space‎, ‎$R = D \propto E$ be the trivial extension of $D$ by $E$‎, ‎and $w$ be the so-called $w$-operation‎. ‎In this paper‎, ‎we show that‎ ‎$R$ is a $w$-FF ring if and only if $D$ is a $w$-FF domain; and‎ ‎in this case‎, ‎each $w$-flat $w$-ideal of $R$ is $w$-invertible.

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Main Subjects

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### History

• Receive Date: 08 August 2016
• Revise Date: 30 January 2017
• Accept Date: 31 January 2017