TIGHT CLOSURE AND STRONG TEST IDEALS

For certain classes of rings we give an affirmative answer to whether there exists a uniform bound on the least power of the tight closure o f an arbitrary ideal which lies in the ideal. The rings need to have t he property that module each minimal prime there exists a resolution o f singularities obtained by blowing up an ideal. In this case we prove the existence of a 'strong' test ideal, and then apply this existence to give an affirmative answer to the uniform bound question. (C) 1997 Elsevier Science B.V.