Lovely little lemmas

Or maybe ‘amusing observations’

Rings are abelian

Posted on 14 November 2018 by Remy

In this post, we prove the following well-known lemma:

Lemma. Let $(R,+,\times,0,1)$ satisfy all axioms of a ring, except possibly the commutativity $a + b = b + a$ . Then $(R,+)$ is abelian.

That is, additive commutativity of a ring is implied by the other axioms.

Proof. By distributivity, we have $2(a+b) = 2a + 2b$ , so multiplication by $2$ is a homomorphism. By our previous post, this implies $R$ is abelian. $\qedsymbol$

One thought on “Rings are abelian”

Sam on 15 November 2018 at 13:27 said:

Hilarious. I didn’t even know this.

Exercise: Find a counterexample for the same statement but for (R, +, •, 0) (i.e., without unit). (It’s actually not hard to find a minimal one, as I know you like to do!)

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One thought on “Rings are abelian”

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