On the Rees algebra and the conductor of an ideal

For an ideal I in a Noetherian ring R, we introduce and study its conductor as a tool to explore the Rees algebra of I. The conductor of I is an ideal C(I)⊂ R obtained from the defining ideals of the Rees algebra and the symmetric algebra of I by a colon operation. Using this concept we investigate when adding an element to an ideal preserves the property of being of linear type. In this regard, a generalization of a result by Valla in terms of the conductor ideal is presented. When the conductor of a graded ideal in a polynomial ring is the graded maximal ideal, a criteria is given for when the Rees algebra and the symmetric algebra have the same Krull dimension. Finally, noting the fact that the conductor of a monomial ideal is a monomial ideal, the conductor of some families of monomial ideals, namely bounded Veronese ideals and edge ideals of graphs, are determined.