Visualizing kinds of rings

When I first saw ring theory, my impression was that there were dozens of kinds of rings with dozens of special relations between them—more than I could keep up with. In reality, there just a few basic kinds of rings, and the relations between them are simple.

Here’s a diagram that shows the basic kinds of rings and the relations between them. (I’m only looking at commutative rings, and I assume ever ring has a multiplicative identity.)

Types of commutative rings

The solid lines are unconditional implications. The dashed line is a conditional implication.

  • Every field is a Euclidean domain.
  • Every Euclidean domain is a principal ideal domain (PID).
  • Every principal ideal domain is a unique factorization domain (UFD).
  • Every unique factorization domain is an integral domain.
  • A finite integral domain is a field.

Incidentally, the diagram has a sort of embedded pun: the implications form a circle, i.e. a ring.

More mathematical diagrams:

2 thoughts on “Visualizing kinds of rings

  1. Actually, the connections start being murky when you get to local rings, Noetherian rings, Dedekind rings, DVRs, Prufer rings and all that. At least, that is when I first got the impression that it can’t be possible to keep track of all the relations. When I first read your introduction, I thought you were going to provide a grand unified theory of that.

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