Welcome to the Database of Ring Theory!

A repository of rings, their properties, and more ring theory stuff.

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See rings and properties on record.

All rings » Commutative rings »

All properties » Commutative properties »


You can search for rings by their properties. If you are only interested in commutative rings, try the specialized search with expanded, commutative-only properties.

All rings » Commutative rings »

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There are a lot of ways you could help, even if you don't know what specifically what yet.

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Recent activity

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  • At Sakurai's recommendation, I've added the $p$-adic integers and their number field. Thanks!
  • Added Ster's ring which can be used to show that "clean" is not a Morita invariant property.
  • Sakurai suggested adding the Akizuki–Hopkins–Levitzki theorem, fill out $F_2[Q_8]$'s properties, and found a few typo corrections. Thanks!
  • Just rolled out what (I hope) are improvements! You should be able to see some of the changes on the search pages, for example.
  • More juicy examples here, here and here. They're all closely related and quite fascinating. The details pages will fill you in!
  • I added and example due to Cohn of a domain that doesn't have IBN.
  • Check out Page's example of a one-sided finitely pseudo-Frobenius ring!
  • I added Hurwitz and Lipshitz quaternion rings, as well as Gaussian integers.
  • I added another classic Bergman example, a right-not-left primitive ring. For a long time I thought it was already entered... Bergman has so many amazing examples!
  • I added Nakayama's example of a QF ring that isn't Frobenius! Can't believe I didn't add it a long time ago...


View a list of resources for studying ring theory.

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  • Math Counterexamples: Mathematical exceptions to the rules or intuition. (Here you can find counterexamples from a variety of mathematical disciplines.)
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  • The Center of Ring Theory and its Applications at Ohio University
  • GroupProps the group properties wiki. It has lots of detailed data on specific groups.
  • The Stacks Project: an open source textbook and reference work on algebraic stacks and the algebraic geometry needed to define them
  • nLab: a wiki-lab for collaborative work on Mathematics, Physics and Philosophy