  Let p be an odd prime, and k_\infty the cyclotomic Z_p-extension of an
abelian field k. For a finite set S of rational primes which does not include
p, we will consider the maximal S-ramified abelian pro-p extension
M_S(k_\infty) over k_\infty. We shall give a formula of the Z_p-rank of
Gal(M_S(k_\infty)/k_\infty). In the proof of this formula, we also show that
M_{q}(k_\infty)/L(k_\infty) is a finite extension for every real abelian field
k and every rational prime q distinct from p, where L(k_\infty) is the maximal
unramified abelian pro-p extension over k_\infty.
