STRONG MORI MODULES OVER AN INTEGRAL DOMAIN

Abstract

Let D be an integral domain with quotient field K, M a torsion-free D-module, X an indeterminate, and <TEX>$N_v=\{f{\in}D[X]|c(f)_v=D\}$</TEX>. Let <TEX>$q(M)=M{\otimes}_D\;K$</TEX> and <TEX>$M_{w_D}$</TEX>={<TEX>$x{\in}q(M)|xJ{\subseteq}M$</TEX> for a nonzero finitely generated ideal J of D with <TEX>$J_v$</TEX> = D}. In this paper, we show that <TEX>$M_{w_D}=M[X]_{N_v}{\cap}q(M)$</TEX> and <TEX>$(M[X])_{w_{D[X]}}{\cap}q(M)[X]=M_{w_D}[X]=M[X]_{N_v}{\cap}q(M)[X]$</TEX>. Using these results, we prove that M is a strong Mori D-module if and only if M[X] is a strong Mori D[X]-module if and only if <TEX>$M[X]_{N_v}$</TEX> is a Noetherian <TEX>$D[X]_{N_v}$</TEX>-module. This is a generalization of the fact that D is a strong Mori domain if and only if D[X] is a strong Mori domain if and only if <TEX>$D[X]_{N_v}$</TEX> is a Noetherian domain.