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    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTime7 days ago

    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).

    diff, v34, current

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