ON SUBMODULE TRANSFORMS T(N) AND S(N)

Abstract

Abstract. In this paper, we give some properties on submoduletransforms. 0. IntroductionLet M be a module over commutative ring Rwith identity, Stheset of nonzero divisors of Rand R S the total quotient ring of R. For anonzero ideal Iof R, let I 1 = fx2R S jxIRg. Iis said to be aninvertible ideal of Rif II 1 = R. Put T= ft2Sjtm= 0 for some m2Mimplies m= 0g. Then Tis a multiplicatively closed subset of Sand ifMis torsion free, then T= S([9,Proposition 1.1]). In particular, if Misa faithful multiplication module then Mis torsion free ([4,Lemma 4.1])and so T= S. So in this case, R T = R S . Let Nbe a submodule of M. Ifx= rt 2R T and n2N, then we say that xn2Mif there exists m2Msuch that tm= rn. Then this is a well de ned operation([9,p399]). Fora submodule N of M, N 1 = fx2R T jxN Mg= [M : R T N]. Wesay that Nis invertible in Mif NN 1 = Mand Mis called a Dedekind(resp. Pruf er) module providing that every nonzero (resp. every nonzero nitely generated) submodule of Mis invertible.Mis called a multiplication module if every submodule Nof Mhasthe form IM for some ideal I of R. An R module M is said to befaithful if Ann (M) = [0 :