STRONG MORI MODULES OVER AN INTEGRAL DOMAIN

Abstract. Let D be an integral domain with quotient field K, M atorsion-free D-module, X an indeterminate, and N v = {f ∈ D[X] | c(f) v = D}. Let q(M) = M ⊗ D K and M w D = {x ∈ q(M) | xJ ⊆ M fora nonzero finitely generated ideal J of D with J v = D}. In this paper,we show that M w D = M[X] N v ∩ q(M) and (M[X]) w [X] ∩ q(M)[X] =M w D [X] = M[X] N v ∩ q(M)[X]. Using these results, we prove that Mis a strong Mori D-module if and only if M[X] is a strong Mori D[X]-module if and only if M[X] N v is a Noetherian D[X] N v -module. This isa generalization of the fact that D is a strong Mori domain if and onlyif D[X] is a strong Mori domain if and only if D[X] N v is a Noetheriandomain. 0. IntroductionLet R be a commutative ring with identity. For any R-module A, letA[X] = {m 0 +m 1 X +···+m k X k | m i ∈ A}be the set of all polynomials in X with coefficients in A. Then A[X] = A ⊗ R R[X]. For all f = m 0 +m 1 X +···+m k X k and g = n 0 +n 1 X +···+n l X l inA[X] with k ≤ l and h = a 0 + a 1 X + ···+ a