Characteristic series of an Iwasawa module (${\mathrm{char}}_{{\mathbb{Z}}_p[[S]]}(M)=(f_M(S))$)

For a finitely generated torsion Iwasawa module, this is a generator of its characteristic ideal, unique up to a unit of the formal power series ring. It may be chosen as a power of $p$ times a product of distinguished polynomials. The series itself and the ideal it generates are different objects; changing a generator by a unit preserves the ideal. When the finite Euler characteristic of a one-variable Iwasawa module is defined, the inverse p-adic absolute value of its constant term gives that invariant.