so I just need a gentle push to help get me started here! I've been combing over the details in my book and I really just can't find a way to show two things are not isomorphic. Let $\mathbb{Z}$ be the integers and $\mathbb{Q}$ be the rationals. I think it should have something to do with identities... but i'm just not sure! Any tips are appreciated.
a) Show $3\mathbb{Z} \ncong 2\mathbb{Z}$
b) Show $\mathbb{Z}[x] \ncong \mathbb{Q}[x]$
c) Find all homomorphic images of $\mathbb{Z}$
Assuming that you have wrongly tagged the question and the structure on these mathematical objects is of ring.
part $(a)$, they are isomorphic as groups as both are infinite cyclic group and upto isomorphism every infinite cyclic group is isomorphic to $\mathbb{Z}$.
Assuming there is a typo, Suppose there is an isomorphism of rings $2\mathbb{Z}$ and $3\mathbb{Z}$ then it would give an isomorphism of the underlying abelian groups and you can show that there are only two such maps. Check whether they satisfy the ring homomorphism properties.
Hint for part $(b)$, Unit of $\mathbb{Z}$ are .... and units of $\mathbb{Q}$ are .....
Hint for part $(c)$ To find all homomorphic image of $Z$ you need to find all the ideals of $Z$ and find all quotient ring.