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Archive for January, 2012

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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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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