Denote by R the real-linear span of e, e,..... e, where e = 1, e-e = + e-e = -2δ,1 ≤ i, j ≤ n. Under the concept of left-monogeneity defined through the generalized Cauchy-Riemann operator we obtain the direct sum decomposition of L (R), n > 1, L(R ) = Σ ⊕ Ω, where Ω is the right-Clifford module of finite linear combinations of functions of the form R(x)h(\x\), where, for d = n + 1, the function R is a k- or -(d + \k\ -2)-homogeneous lefl-monogenic function, for k > 0 ork < 0, respectively, and h is a function defined in [0, ∞) satisfying a certain integrability condition in relation to k, the spaces Ω are invariant under Fourier transformation. This extends the classical result for n = 1. We also deduce explicit Fourier transform formulas for functions of the form R(x)h(r) refining Bochner's fonnula for spherical k-harmonics.