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    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 13th 2020

    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

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 12th 2021

    Touched the wording and formatting of the Idea-section, in an attempt to streamline a little.

    diff, v35, current

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