# Invariants of $\mathbb{Z}$-module isomorphisms

Let $$\theta = \sqrt[3]{2}$$, and $$M = \{4, \theta, \theta^2\}$$, $$N = \{1, 2\theta, 2\theta^2\}$$ be finitely generated $$\mathbb{Z}$$-modules in $$\mathbb{Q}(\theta)$$. I would like to show either $$M \cong N$$ (which seems to be false) or $$M \not\cong N$$, but I do not know how to get started. Can anyone help, maybe by providing me with some invariants of isomorphism of this sort? Thanks in advance!

edit: I am sorry for the poor quality of my question. Thanks to Mr. julio_es_sui_glace, whose answer has solved my question. Still I would like to provide some contexts so that the description may be improved.

I comes up with the problem when reading 2.2, Chapter 2 of Borevich & Shafarevich's Number Theory. Let $$M$$ denote a finite-generated, torsion-free and full $$\mathbb{Z}$$-module in an algebraic number field, $$\mathfrak{O}_M$$ its coefficient ring. The book constructs first a injection $$\mathfrak{O}_M \to M$$, and shows that they are of the same rank. I thought at that time this was sufficient for $$\mathfrak{O}_M \cong M$$ (module isomorphism). But later on the book mentions another injective isomorphism $$M \to \mathfrak{O}_M$$, which made me doubt if $$\mathfrak{O}_M \cong M$$. To understand the necessity of building the latter isomorphism, I tried to calculate some examples, and (as glace said) confused module isomorphism with ring isomorphism.

Now I guess the book constructs the latter isomorphism because it is explicit (by multiplying $$M$$ with a certain natural number), showing not only the coincidence of module structure, but also that $$M$$ is similar to a module in $$\mathfrak{O}_M$$ which is used in further developments (e.g. showing that $$M$$ contains only a finite number ofpairwise-nonassociate elements with given norm).

I should thank those who give me advice to improve the question. I also hope there will be answers improving my understanding on the intention of the two isomorphism constructed in the book.

You have a clear isomorphism of modules (but not rings) given by any bijection between $$\{4,\theta,\theta^2\}$$ and $$\{1,2\theta,2\theta^2\}$$.
This is clear since these modules are free since both $$\{4,\theta,\theta^2\}$$ and $$\{1,2\theta,2\theta^2\}$$ are basis of the $$\mathbb{Q}$$-vector space $$\mathbb{Q}[\theta]$$.
e.g. you have the isomorphism of modules given by $$\begin{array}{rcl} M & \longrightarrow & N\\ 4 & \longmapsto& 1 \\ \theta & \longmapsto& 2\theta\\ \theta^2 & \longmapsto& 2\theta^2 \end{array}$$