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.

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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.

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Site update: Three new features!
On ring detail pages you will now find Dimensions and Subsets tabs. Take a look at $\mathbb Z$ to see how they work. Also, there is now a ring theory literature errata page so we can track mistakes as we find them. You can get to it through the Citations menu item.
new property: PCI rings
I added PCI rings to the collection. Interestingly, this is the only property I'm aware of in the database for which it is currently unknown whether or not the definition is symmetric or asymmetric.
rings improved: Thanks KMattis!
As a result of two or three suggestions left by KMattis, a total of around 100 properties on a handful of rings were decided. Thanks! A good example of how sometimes just a few suggestions can snowball into an avalanche.
new ring: Right-not-left V ring
From Michler and Villemayor's classic paper, I've added the example they gave to prove asymmetry of V rings.
Improved ring: Updated ring 50
Sakurai made a single suggestion leading to resolution of 30 properties on $R_{50}$!
new rings/new properties: Hutchins' book
Currently I'm skimming through Hutchins' book (see the bibliography section) trying to make quick additions. Not sure if we'll ever absorb all the examples listed there (180 with pseudo repetition, plus allusions to more).
site news: On-site notifications
For those of you with accounts, you'll now see unread message notifications. Also, your profile page looks a lot better.
new ring: $p$-adic numbers
At Sakurai's recommendation, I've added the $p$-adic integers and their number field. Thanks!
new ring: Added Ster's example disproving Morita invariance of "clean"
Added Ster's ring which can be used to show that "clean" is not a Morita invariant property.
new theorem: Thanks Sakurai
Sakurai suggested adding the Akizuki–Hopkins–Levitzki theorem, fill out $F_2[Q_8]$'s properties, and found a few typo corrections. Thanks!


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