Pure projective and pure injective modules over polynomial rings
3
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
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.
Abstract
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.