I started writing this post because, for whatever reason, I keep forgetting what the difference is between a ring and a group, which is funny to me because I never forget the difference between a semiring and a semigroup – although other people do, because it’s quite easy to forget! So, I wanted a fast reference to the kinds of algebraic structures that I am most often dealing with in one way or another, usually because I’m writing Haskell (which has some reliance on terminology and structure from abstract algebra and category theory) or I’m trying to read a book about category theory and they keep talking about “groups.” Wikipedia, of course, defines all these structures, and that’s fine, but what I need in those times is more of a refresher than an in-depth explanation.