Bulletin of the
Korean Mathematical Society BKMS

We study the purity of modules over polynomial rings. It is proven that (1) $X$ is a pure projective left $R$-module if and only if $\textbf{T}(X)$ is a pure projective left $R[x]$-module if and only if $\textbf{Z}(X)$ is a pure projective left $R[x]$-module; (2) $f: A\rightarrow N$ is a pure projective precover (resp. pure projective cover) in $R$-Mod if and only if $\textbf{Z}(f): \textbf{Z}(A)\rightarrow \textbf{Z}(N)$ is a pure projective precover (resp. pure projective cover) in $R[x]$-Mod. Also, we obtain dual results about pure injective modules.

Keywords: Polynomial ring; pure projective module; pure injective module.

MSC numbers: 16D40; 16D50; 18G25.