Inverse polynomial modules induced by an $R$-linear map
Bull. Korean Math. Soc. 2010 Vol. 47, No. 4, 693-699
Published online July 1, 2010
Sangwon Park and Jinsun Jeong
Dong-A University, Dong-A University
Abstract : In this paper we show that the flat property of a left $R$-module does not imply (carry over) to the corresponding inverse polynomial module. Then we define an induced inverse polynomial module as an $R[x]$-module, i.e., given an $R$-linear map $f:M \rightarrow N $ of left $R$-modules, we define $N+x^{-1}M[x^{-1}]$ as a left $R[x]$-module. Given an exact sequence of left $R$-modules $$ 0 \longrightarrow N \longrightarrow E^{0} \longrightarrow E^{1}\longrightarrow 0, $$ where $E^{0}, E^{1}$ injective, we show $E^{1}+x^{-1}E^{0}[[x^{-1}]]$ is not an injective left $R[x]$-module, while $E^{0}[[x^{-1}]]$ is an injective left $R[x]$-module. Make a left $R$-module N as a left $R[x]$-module by $xN=0$. We show $$ {\rm inj}\dim_{R}N=n \hskip 0.5cm \hbox{implies} \hskip 0.5cm {\rm inj}\dim_{R[x]}N=n+1 $$ by using the induced inverse polynomial modules and their properties.
Keywords : flat module, injective module, inverse polynomial module, induced module
MSC numbers : Primary 16E30; Secondary 13C11, 16D80
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