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...)