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nLab > Latest Changes: nonunital ring

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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 4th 2023

    Added:

    A survey of various notions between unital rings and nonunital rings:

    • Patrik Nystedt, A survey of s-unital and locally unital rings, Revista Integración, temas de matemáticas Escuela de Matemáticas, Universidad Industrial de Santander, Vol. 37, N◦ 2, 2019, pág. 251–260. doi.

    diff, v21, current

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeSep 4th 2023
    • (edited Sep 4th 2023)

    I do not quite understand why unitisation is treated for commutative rings here. In my recollections, formally adding a unit is useful when treating module categories over general (=nonunital) rings as categories of unital modules over associated unital rings. (Of course, the category of nonunital modules over the associated unital ring are different.)

    However there are some important differences between unital and nonunital rings. First of all, in the unital case every Artinian ring is Noetherian, but not other way around. This is a bit misterious at a first glance, but the true reason of breaking the symmetry is the existence of the unit element: the fact does not extend to the nonunital context.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 5th 2023

    I don’t see that removing the word “commutative” throughout would break any of the propositions (esp. Propositions 3.6 and 3.8), so I think it would be fine to remove it. My guess is that whoever wrote it in was simply more familiar with the commutative case.

  4. changed higher algebra - contents to algebra - contents in context sidebar

    Anonymouse

    diff, v23, current

1 to 4 of 4

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