Examples of Groups (resp. Rings, Fields, etc.) Which Are Isomorphic to a Proper Subgroup (resp. Subring, Subfield, etc.)

Question

Was just reading this question When is a group isomorphic to a proper subgroup of itself? and was wondering not about the conditions for being isomorphic to a proper subgroup, subring, etc., but about examples of these things happening. (Certainly one necessary condition is that our algebraic object be infinite as a set. Otherwise we cannot have a bijection between the initial set and a proper subset.)

I believe one example was given in the above link using even powers of a polynomial ring $R[x]$ and the ring $R[x]$ itself. I believe another example is in Dummit and Foote with the roots of unity or something like this. Anyway, have at it!

(Here is another related post: Rings with isomorphic proper subrings)

(Feel free to also post answers with maybe manifolds which are homeomorphic/diffeomorphic/biholomorphic to proper submanifolds or things like that if you have some favorites!)

(The only category which I know excludes this business is algebraic geometry...but only kinda if you deal with incomplete intersections...)

Answer 1

$\mathbb{Z} \cong n \mathbb{Z}$ by way of the group isomorphism defined by $\varphi (k) = nk$, with $k,n \in \mathbb{Z}$ of course.

Answer 2

If you have an example for fields, you have an example for rings and abelian groups at the same time:

Take the field of rational polynomials $F(x)$ where $F$ is a field. The map $x\mapsto x^2$ defines a ring homomorphism of $F(x)\to F(x)$ which is necessarily injective (since $F(x)$ is a field) but not onto (its image is $F(x^2)$. The image is an isomorphic copy of $F(x)$ strictly contained in $F(x)$.

Of course, this means there is an infinite strictly descending chain of isomorphic copies...