Welcome to the Database of Ring Theory!

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

Learn more »


See rings and properties on record.

All rings » Commutative rings »

All properties » Commutative properties »

By dimension »


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 »

By keyword »


View a list of resources for studying ring theory.

Start learning »


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

Make the site better »

Recent activity

Site news RSS »

new ring: Right-not-left free ideal ring
new ring: the Toeplitz-Jacobson algebra
Thanks dbossaller for the excellent example of a ring which has IBN but is not Dedekind finite.
New ring: Cozzens' simple left PID
This example due to Cozzens is often cited as showing how much asymmetry is possible in simple rings.
new theorem: Did you know...
Commutative von Neumann regular rings are characterized by the fact that their polynomial ring over a single variable is semihereditary? That is the content of this theorem of Camillo.
new ring: Almost-but-not-quite-Dedekind
Added an example of an almost Dedekind ring which isn't Dedekind. In particular it has Krull dimension 1, all of its localizations are Noetherian, but it itself is not Noetherian.
New ring: asymmetry of PF
This is a ring which is one-sided pseudo-Frobenius. The construction is involved, and not complete yet. Also, if you had any trouble visiting the theorems section recently, please try again. I fixed a bug there.
new ring: Asymmetry of principally injective
This is an example of a ring which is one-sided principally injective.
New rings: V domains
Added this ring and this ring, examples of simple right V domains, which I hear are pretty rare.
Site news: Ring maps
I posted a few graphs of implications between ring properties. This is the first time I've regenerated the graphs in a long time, so there are probably problems. Please let me know if you find any. Thanks!
site news: 100th ring winner (and more!)
Congratulations MatheinBoulomenos for the winning submission! The next several rings are submissions from the contest too, so check them out. Also thanks to kc_kennylau for working so hard filling in a lot of gaps, especially around ring cardinalities.

Support us

Buy me a coffeeBuy me a coffee!

Other helpful sites

If you like this site, then you also might like these sites: