On tamely ramified Iwasawa modules for the cyclotomic Zₚ-extension of abelian fields

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Let p be an odd prime, and k∞ the cyclotomic Zₚ-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 Mₛ(k∞) over k∞. We shall give a formula of the Zₚ-rank of Gal(Mₛ(k∞)/k∞). In the proof of this formula, we also show that Mq(k∞)/L(k∞) is a finite extension for every real abelian field k and every rational prime q distinct from p, where L(k∞) is the maximal unramified abelian pro-p extension over k∞.