Efficient generation of ideals in overrings of polynomial rings

Let R   be a commutative Noetherian ring. This paper examines the question of efficient generation of ideals in a birational overring of R[X]R[X]. Also, we study similar questions for rings of the type R[X,Y]/(XY)R[X,Y]/(XY). These studies enable us to compute the Euler class groups for those two types of rings.