The Euler class group of a Noetherian ring

For a commutative Noetherian ring A of finite Krull dimension containing th e field of rational numbers, an Abelian group called the Euler class group is defined. An element of this group is attached to a projective A-module o f rank = dim A and it is shown that the vanishing of this element is necess ary and sufficient for P to split off a free summand of rank 1. As one of t he applications of this result, it is shown that for any n-dimensional real affine domain, a projective module of rank n (with trivial determinant), a ll of whose generic sections have n generated vanishing ideals, necessarily splits off a free direct summand of rank 1.