Iwasawa-module rank (${\mathrm{rank}}_{\Lambda }M={\dim }_{\mathrm{Frac}\Lambda }(M{\otimes }_{\Lambda }\mathrm{Frac}\Lambda )$)

For a finitely generated module over the integral domain $\Lambda$, its Iwasawa-module rank counts free summands up to pseudo-isomorphism. Rank zero is equivalent to being a torsion module. A positive rank $r$ forces the ${\mathbb{Z}}_p$-ranks of finite-layer coinvariant modules to grow like $r{p}^n+O(1)$.