First order conditions are given which are necessary for a commutative regular ring to have a prime integrally closed extension. If the ring is countable these conditions are also sufficient. In [8] an example was given of a commutative regular ring with no prime model extension to a commutative integrally closed regular ring. In this paper we give (in §2) first order conditions which are neces...

The main result of this paper is the explicit determination of the core of integrally closed ideals in two-dimensional regular local rings. The core of an ideal I in a ring R was introduced by Judith Sally in the late 1980's and alluded to in Rees and Sally's paper RS]. Recall that a reduction of I is any ideal J for which there exists an integer n such that JI n = I n+1 NR]. In other words, J ...

We give conditions for a maximal divisorial ideal to be t-maximal and show with examples that, even in a completely integrally closed domain, maximal divisorial ideals need not be t-maximal.

This paper studies the Ratliff-Rush closure and the integral closure of an ideal in a valuation and Prüfer domains. By definition, the RatliffRush closure (respectively the integral closure) of an ideal I of a domain R is the ideal Given by Ĩ := S (I :R I ) (respectively I := {x ∈ R|x satisfies an equation of the form x + a1x + · · ·+ ak = 0, where ai ∈ I i for each i ∈ {1, . . . , k}}). An ide...

This paper studies the Ratliff-Rush closure and the integral closure of an ideal in a valuation and Prüfer domains. By definition, the RatliffRush closure (respectively the integral closure) of an ideal I of a domain R is the ideal given by Ĩ := S (I :R I ) (respectively I := {x ∈ R|x satisfies an equation of the form x + a1x + · · ·+ ak = 0, where ai ∈ I i for each i ∈ {1, . . . , k}}). An ide...