Pseudo-isomorphism in Iwasawa Theory Let $\Lambda$ denote the Iwasawa algebra $\mathbb{Z}_p[[\Gamma]]$, where $\Gamma$ is a group isomorphic to $\mathbb{Z}_p$. We know that $\Lambda$ is homeomorphic to $\mathbb{Z}_p[[T]]$, ring of power series in the variable $T$. Two modules $\Lambda$-modules $A$ and $B$ are said to be pseudo-isomorphic if there is a $\Lambda$-homomorphism $f\colon A\to B$ with finite kernel and cokernel, i.e. they sit in an exact sequence like $$0\to F_1\to A\to B\to F_2\to 0,$$ with $F_1$ and $F_2$ finite $\Lambda$-modules. We write $A\sim B$. If $A$ is finitely generated, then $$A\sim \Lambda^{n}\oplus \bigoplus_i \Lambda/f_i\Lambda$$ for some $f_i\in \Lambda$. The characteristic ideal is $\prod_i f_i\Lambda$. 
I have a few questions:
1) Some authors define the characteristic ideal to be 0 in the non-torsion case, others does not specify. Is there a reason for which one is preferable? Is true in every case that the characteristic ideal well-behaves under exact sequences?
2) The pseudo-isomorphism relation is not symmetric in general, but it is in the case of torsion modules. Is there a simple way to see this?
3)Is this relation transitive in general, or maybe just in the case of torsion? No book says something like this, but if it is not symmetric, how can we say that if $A$ and $B$ are pseudo-isomorphic, then they have the same characteristic ideal?
For (3), it seems to me natural that if $f\colon A\to B$ and $g\colon B\to C$ have finite kernel and cokernel, also the composition has finite kernel and cokernel. For example, the kernel of the composition is the set of elements going to zero under $f$, plus the ones going to the kernel of $g$, and also this second part is finite since both kernels are finite. Am I missing something?
 A: 1) Characteristic ideal. There is no need to define anything like a characteristic ideal (or polynomial) for a free $\Lambda$-module for the simple reason that there is no simpler object than a free object (in a relevant category). For a noetherian torsion $\Lambda$-module $M$, the utility of the ch. s. is that it somehow plays the role  of the determinant in linear algebra. However the genuine deep meaning, for certain $M$'s coming from arithmetic (such as the projective limit of the p-class groups when climbing up the p-cyclotomic tower) is that the ch. s. is intimately related to an adequate p-adic zeta function ("Main Conjecture" of Iw. theory). Given a short exact sequence of torsion $\Lambda$-modules, the ch. ideal of the middle term is the product of those of the extreme terms.
2) Symmetry. It is not so straihgtforward to show that pseudo-isomorphim between two torsion $\Lambda$-modules is a symmetric relation. The fastest proof I know uses the uniqueness of the decompostion up to pseudo-isomorphism of such a module into a direct sum of modules of the form $\Lambda /P^n$, where $P$ is a prime ideal of height 1.This can be done by hand, but you can also consult Serre's Bourbaki talk, "Classes des corps cyclotomiques"'1960), lemma 5 and thm.7. 
3) Transtivity is easily reduced to the case of torsion modules. Then the approach is the same as in 2).
