in this paper, we give a generalization of the integral dependence from rings to modules. we study the stability of the integral closure with respect to various module theoretic constructions. moreover, we introduce the notion of integral extension of a module and prove the lying over, going up and going down theorems for modules.

Let D be an integral domain with identity having quotient field K. This paper gives necessary and sufficient conditions on D in order that each integrally closed subring of O should belong to some subclass of the class of integrally closed domains ; some of the subclasses considered are the completely integrally closed domains, Prüfer domains, and Dedekind domains. 1. The class of integrally cl...

Let (A, m) be an excellent normal local ring with algebraically closed residue class field. Given integrally closed m-primary ideals I ⊃ J , we show that there is a composition series between I and J , by integrally closed ideals only. Also we show that any given integrally closed m-primary ideal I, the family of integrally closed ideals J ⊂ I, lA(I/J) = 1 forms an algebraic variety with dimens...

We show that certain two-dimensional, integrally closed monomial modules can be uniquely written as a countable product of isomorphic copies of simple integrally closed monomial ideals.

R. P. Stanley proved the Upper Bound Conjecture in 1975. We imitate his proof for the Ehrhart rings. We give some upper bounds for the volume of integrally closed lattice polytopes. We derive some inequalities for the delta-vector of integrally closed lattice polytopes. Finally we apply our results for reflexive integrally closed and order polytopes.