In this post, I’d like to record a few basic definitions and results regarding noncommutative rings. This is a subject clearly of great importance and generality, but I haven’t had much exposure to it, and I’m trying to fix that. I am working mostly from Lam’s *A first course in noncommutative rings*.

## Archive for January, 2012

## A first blog post on noncommutative rings

Posted in module theory, ring theory, tagged enriched categories, semisimplicity on January 25, 2012 | 3 Comments »

## A less biased definition of a group

Posted in category theory, group theory, tagged abstract nonsense, Lawvere theories on January 16, 2012 | 23 Comments »

Here’s what seems like a silly question: what’s the universal group? That is, what’s the universal example of a set together with maps

satisfying the identities

- ,
- ,
- ?

A moment’s reflection shows that there isn’t such a group; the existence of the groups , where is an arbitrary set, shows that there exist groups of arbitrarily large cardinality, so no particular group can be universal.