Here is the hierarchy of algebraic objects ranging from semigroups to fields. Each ellipse contains an example of its corresponding object strictly in its respective classification.
If you know good examples that you think are worthy and would like to see on this picture please let me know and I'd be happy to include it.
*Unfortunately, this picture can technically be misleading around the "Ring" and "Abelian Group" portion as it turns out every abelian group can have a ring structure attached to it if the law of composition is defined properly, it just might not be unital. For example, you can define multiplication to send everything to the additive identity. There are clearly no units, however, this will still meet all of the requirements to be a ring (depending on your definition of a ring). You can read more about it here and here. Some people like to use the term rng, pronounced "rung", to distinguish between rings without unit and rings with. Personally, I believe that nonunital rings should be categorized as rings and the almighty Hungerford, scripture for algebraists, doesn't require rings to be unital so therefore it must be true. That being said, I still decided to have separate classifications to distinguish the differences that we typically care about between the two algebraic objects.