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Moving discussion here and summarizing content in the text
+– {: .query} Mike: Why only rings without units (that is, rngs)? Intuitively, what important properties do the above listed examples share that are not shared by rings with units?
Zoran Skoda: I want to know the answer as well. It might be something in the self-dual axioms. For unital rings artinian implies noetherian but not other way around; though the definitions of the two notions are dual.
Toby: The category of unital rings and unitary ring homomorphisms has no zero object.
Mike: Ah, right. Is it protomodular? I think I will understand this definition better from some non-examples that violate each clause individually.
walt: It is protomodular. This follows from the main theorem of Characterization of Protomodular Varieties of Universal Algebra by Bourn and Janelidze. By that theorem any variety that contains a group will be protomodular. Unital rings only fail to be semiabelian for the trivial reason that ideals aren’t subrings.
=–
Maybe the result on protomodularity (with citation) mentioned by walt citing Bourn and Janelidze should be moved to CRing (and also Ring, if it holds for non-commutative rings).
removing old query box:
+– {: .query} Urs: how can I understand that this (has to?) involve the opposite category?
Mike: Well, as the previous example shows, $Set_*$ itself is not semi-abelian. The way I’m thinking of it is that a surjection of pointed sets is not determined by its kernel, but an injection of pointed sets is determined by its cokernel. =–
Anonymous
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