The plus closure of an ideal

If R is a local integral domain let R+ denote the integral closure of R in an algebraic closure of its quotient field. If z R+, we would like to understand the conditions under which z IR+, where I is an ideal of R. Necessary and sufficient conditions on the coefficients of the minimal irreducible polynomial for z are known when I is generated by two elements of a regular system of parameters and when z is in a degree two extension of R. In this article we obtain results for the case when z3 R, as well as a sufficient condition for z to be in IR+ when za R for a 1 and I has a finite number of generators.