Let R be a Noetherian commutative ring of dimension n > 2 and let A = R [ T , T −1 ] . Assume that the height of the Jacobson radical of R is at least 2. Let P be a projective A-module of rank n = dim A − 1 with trivial determinant. We define an abelian group called the “Euler class group of A,” denoted by E ( A ) . Let χ be an isomorphism from A to det ( P ) . To the pair ( P , χ ) , we associate an element of E ( A ) , called the Euler class of P, denoted by e ( P , χ ) . Then we prove that a necessary and sufficient condition for P to have a unimodular element is the vanishing of e ( P , χ ) in E ( A ) . Earlier, Bhatwadekar and Raja Sridharan have defined the Euler class group of R, denoted by E ( R ) , and have proved similar results for projective R-module of rank n. Later, M.K. Das defined the Euler class group of the polynomial ring R [ T ] , denoted by E ( R [ T ] ) , and proved similar results for projective R [ T ] -modules of rank n with trivial determinant.