Last time we talked about sets which support a single binary operator. That's all well and good, but things get really interesting once we have two of those operators. Thus, for lack of a better name, I present a map of ring theory. The majority of this isn't actually part of ring theory proper, of course; but that's as good a name as any, since we'll want to distinguish these from lattice theory which also has two interlocking operators.
In the center of the map are (unital) rings. For those who aren't aware, in older literature, what are now called pseudorings used to be called "rings", and what are now called rings used to be called "unital rings" or "rings with unit". I've run into very few proper pseudorings of interest, so I support this change in terminology. In any case, if we start from rings and move upward we get rid of properties. In addition to pseudorings we also get semirings, which are utterly ubiquitous. In the next post I'll give a few dozen of the most common semirings, so I'll save my rant for then. If we keep moving up then we start getting rid of associativities and distributivities, resulting in things like seminearrings (aka near semirings) which arise naturally in certain kinds of parsing problems. This area is poorly explored, hence the dearth of names in the upper part of the map. This is, however, where I've been spending a lot of time lately; so you'll probably hear more about seminearrings and their ilk in the near future. As the names suggest, we have both left-near and right-near versions of these various structures, though I only draw the right-near ones for clarity.
Moving downward from semirings there are two directions to head. Off to the left we run into Kleene algebras and lattice theory yet again. And to the south we run into the swamp along the way to fields. In spite of their apparent axiomatization, fields are not really algebraic objects, which is a big part of the reason for all this mess. In the lower left we see a chain of inclusions based on some peculiar properties like every ideal being principal, the existence of greatest common divisors (though not necessarily computable with Euclid's algorithm), the ascending chain condition on principal ideals, etc. These properties will be familiar to the actual ring theorists, as would numerous other properties I didn't bother putting on here. Off to the lower right we get a different sort of breakdown. In particular, before we get unique inverses for all non-zero elements, we can instead just have pseudoinverses or strong pseudoinverses. This is similar to the distinction between (von Neumann) regular semigroups vs inverse semigroups.
There's plenty more to say about ring theory and related areas, but that would be a whole series of posts on its own. This is actually the first map I started to make, because this is the region where we find so many mathematicians coming in from different areas and not being familiar with all that has been done before. As I'm sure you notice, quite a lot has been done, both in breadth and in depth. I brought this map up once when chatting with Amr Sabry, and he's the one who convinced me to finally get around to posting all these. So, hopefully these'll give y'all a better lay of the land.
There are some notable omissions from this map as it stands. In particular, complete semirings (aka *-semirings) are missing, as are rings with involution (aka *-rings), as are the various constructive notions of fields (like discrete fields, Heyting fields, residue fields, meadows, etc.). Next time I'll talk a bit about complete semirings; they were omitted mainly for lack of space, but should be included in future versions. The various constructive notions of fields were also omitted mainly for space reasons. I'll probably end up making a close-up map of the swamplands between rings and fields in order to do justice to them. Rings with involution were omitted mainly because I'm not entirely sure where the best place to put them is. As the name suggests, they've been primarily studied in contexts where you have a full-blown ring. However, there's nothing about the involution which really requires the existence of negation. I've started chatting with Jacques Carette about related topics recently, though; so perhaps I'll have more to say about them later.
This map is released under Creative Commons Attribution-ShareAlike 3.0. Any questions? Anything I missed? Anything that needs further explanation?