Consider a ring $R$ and two ideals $I,J$ such that $I\subseteq J$. There is a natural projection $R/I\to R/J$ by the isomorphism theorem. There is, in general, no homomorphism going the other way around (there are functions as axiom of choice gives right inverses for surjective functions; see this post).
However, in these lecture notes in at least two proofs (p. $45$ bottom, lemma $4.7$; p. $47$ top, lemma $4.10$) the following "fact" seems to be employed:
Let $R$ be a ring and $I,J$ ideals such that $I\subseteq J$. Then $R/J\subseteq R/I$.
This fact is applied to $R=\mathbb Z_p[[T]]$ the Iwasawa-algebra (it is, in fact, $\mathcal O_K[[T]]$ for the ring of integers $\mathcal O_K$ of a number field $K$ but I do not think that matters structurally speaking) with $I$ and $J$ (some specific) finitely generated ideals. As this already fails in $\mathbb Z$ (consider $I=\langle0\rangle$ and $J=\langle2\rangle$) which is a very well-behaved ring I am not sure what happens here.
Could someone please explaing what the underlying idea here is? Although it would already help to understand the specific incarnations of the "fact".
Thanks in advance!
EDIT: After some more thinking, two further comments:
- Ignoring the structure there is a natural way of constructing an injective set-function $R/I\to R/J$ (as mentioned above and in the linked Math.SE question). This suffices for the usage in the linked lecture notes as there only something about the cardinality is deduced.
- As my example for $R=\mathbb Z$ illustrated it might be crucial to exclude the zero ideal (as for now I do not see any problems with the unit ideal). At first glance this fixes the problem for $R=\mathbb Z$.
Yes, the underlying idea really is that any set-theoretical surjection $A\to B$ induces a set-theoretical injection $B\to A$ which -- in terms of cardinalities -- gives $|B|\le|A|$. This point of view works out just fine for any application within the notes (and as well within the main source "L.C. Washington: Introduction to Cylcotomic Fields").