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

See rings and properties on record.

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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.

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

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!

View a list of resources for studying ring theory.

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- The Center of Ring Theory and its Applications at Ohio University
- GroupProps: the group properties wiki.
- 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