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For a finitely generated torsion Iwasawa module, its characteristic ideal is the product of the height-one prime ideals raised to the lengths of the corresponding localizations. In the elementary-divisor description it is generated by $p^{\sum a_i}\prod f_j^{b_j}$. It is unchanged by finite modules and multiplicative in short exact sequences of torsion modules.

The invariants $\mu ,\lambda$ of a finitely generated torsion Iwasawa module measure its $p$-power elementary factors and distinguished-polynomial factors. Vanishing of $\mu$ means that the module is finitely generated over ${\mathbb{Z}}_p$, up to finite error. It does not mean that the module is finite: a factor $\Lambda /(T)$ has $\mu =0$ and is ${\mathbb{Z}}_p$.

The classical Iwasawa main conjecture, now a theorem over $\mathbb{Q}$, identifies characteristic ideals of appropriate cyclotomic Iwasawa modules with ideals generated by p-adic L-function power series. For a nontrivial even character in the p-ramified convention, ${\mathrm{char}}_{\Lambda }{X}_{\chi }=(g_{\chi })$ with $g_{\chi }((1+p{)}^{1-s}-1)=L_p(\chi ,s)$. The Kummer reflection in Iwasawa theory accounts for the reflected odd class-group convention.

A homomorphism between finitely generated one-variable Iwasawa modules is a pseudo-isomorphism if its kernel and cokernel are finite. Finite errors do not change the Iwasawa-module rank or the characteristic ideal of a torsion module. They can still change its integral structure, so a pseudo-isomorphism is weaker than an isomorphism.

The Tate twist $M(1)$ multiplies a Galois representation's action by the cyclotomic character. Here ${\mathbb{Z}}_p(1)$ is the inverse limit of $p$-power roots of unity under power maps. For a one-variable Iwasawa module, twisting changes characteristic power series by the corresponding change in the generator's action.

For a Zp-extension, let $L_{\infty }$ be the maximal unramified abelian pro-$p$ extension of $F_{\infty }$. Its Galois group is a compact Iwasawa module, identifiable by Artin reciprocity with the inverse limit of the $p$-primary ideal class groups of finite layers under norms.