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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2011

    for completeness: unitization

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJul 25th 2011

    New entry unital category in the sense of Bourn. Unfortunately not much in accordance to other unitalities (in higher category theory).

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeJul 25th 2011

    I generalised unitalisation to nonassociative algebras.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2011

    Er, wait. I spent some time with renaming “unitalization” to “unitization”. You revereted it? Well, it’s fine with me if that’s more correct!

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 27th 2011

    I prefer “unitalization” myself. (Cf. “abelianization”.)

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeJul 27th 2011

    Sorry Urs, I thought that “unitization” was a typo.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 27th 2011

    What happened was that first I had created the entry titled “unitalization”. Then I was going through some literature and saw it as “unitization”. So I renamed entries and everything. But never mind, you decide on what it should be called! :-)

    • CommentRowNumber8.
    • CommentAuthorjesse
    • CommentTimeSep 21st 2016
    • (edited Sep 21st 2016)

    I added a couple of observations (Remarks 3.3 and 3.4) at the section unitization of the page nonunital ring:

    i) I always thought the just-so definition in terms of A(A)A \mapsto \left(A \oplus \mathbb{Z}\right) with the multiplication (a 1,z 1)(a 2,z 2)(a 1a 2+a 1z 2+z 1a 2,z 1z 2)(a_1, z_1) \cdot (a_2, z_2) \mapsto (a_1 a_2 + a_1 z_2 + z_1 a_2, z_1 z_2) was a bit mysterious, so I pointed out that you can get the same thing by taking A[x]A[x] and quotienting out by relations which say that xx is a unit.

    ii) The universal property of the unitalization and the fact that endomorphism rings of abelian groups are unital means that a module over a nonunital ring is the same thing as a module over its unitalization.