On the kernel and image of the spin representation

Let E be a commutative ring with unit element 1R denoted 1, and such that 2#0, let M be a R-module (such that 1.x = x for all x in M), f: M×M → Ra symmetric bilinear form, C(M, f) the Clifford algebra of the pair (M, f), [3], Ф: Spin(M, f) → HomR(M, M) the well known representation of the spin group Spin(M, f).