It follows from the Weierstrass preparation theorem for formal power series over a complete local ring that the prime ideals of this ring are as follows: The rank of a finitely generated module is the number of times the module Zp[[T]] occurs in it.
This is well-defined and is additive for short exact sequences of finitely-generated modules.
For any finitely generated torsion module there is a quasi-isomorphism to a finite sum of modules of the form Zp[[T]]/(fn) where f is a generator of a height 1 prime ideal.
This invariant is additive on short exact sequences of finitely generated torsion modules (though it is not additive on short exact sequences of finitely generated modules).
The λ-invariant is the sum of the degrees of the distinguished polynomials that occur.
In other words, if the module is pseudo-isomorphic to where the fj are distinguished polynomials, then and In terms of the characteristic power series, the μ-invariant is the minimum of the (p-adic) valuations of the coefficients and the λ-invariant is the power of T at which that minimum first occurs.
Such modules are Artinian and have a well defined length, which is finite and additive on short exact sequences.
In particular this applies to the case when en is the largest power of p dividing the order of the ideal class group of the cyclotomic field generated by the roots of unity of order pn+1.
More general Iwasawa algebras are of the form where G is a compact p-adic Lie group.