Substructure of $\mathbb Q$ that is not a field We have a  structure: $\mathbb Q=\langle \mathbb Q,\,+,\,-,\,*,\,0,\,1 \rangle$

Question: Does $\mathbb Q$ contain a substructure which is not a field?

I have in my notes that there is such a substructure and it is $\mathbb Z$ because $\mathbb Z \subseteq \mathbb Q$

But I don't think that I fully understand it. 

We wrote the axioms of field and that 

$x \neq 0 \rightarrow (\exists y)(x*y=y*x=1)$  does not satisfy rest of the theory.

but I don't know exactly why.
The answer to your question depends on what kind of substructure you're looking for. (That might have been explained in class but didn't get to your notes.)

There is no subset of $\mathbb{Q}$ that is itself a field with the usual addition and multiplication.

$\mathbb{Z}$ is a subring of $\mathbb{Q}$ when you think of the latter as a ring.

Any subset of $\mathbb{Q}$ is a "substructure" if "being a set" is the only structure you care about.
A possible "substructure" here is "subring" in this context. The integers are a subring of the field $\mathbb{Q}$. But they are not a subfield, because not all nonzero elements are invertible, e.g. the equation $2x=1$ has no solution in the integers.