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## A first blog post on noncommutative rings

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.

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## A less biased definition of a group

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

$\displaystyle e : 1 \to G, m : G \times G \to G, i : G \to G$

satisfying the identities

1. $m(e, x) = m(x, e) = x$,
2. $m(x, i(x)) = m(i(x), x)$,
3. $m(x, m(y, z)) = m(m(x, y), z)$?

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

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