 p-ramified Iwasawa module

Let $M_{\infty }$ be the maximal abelian pro-$p$ extension of $F_{\infty }$ unramified away from primes over $p$. Its Galois group is the p-ramified Iwasawa module. In the cyclotomic tower of $\mathbb{Q}({\mu }_p)$ for odd $p$, its Iwasawa-module rank is $(p-1)/2$. It is therefore quite different from the torsion unramified Iwasawa module.