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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeSep 4th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: 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](http://dx.doi.org/10.18273/revint.v37n2-2019003). <a href="https://ncatlab.org/nlab/revision/diff/nonunital+ring/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/nonunital+ring/21">v21</a>, <a href="https://ncatlab.org/nlab/show/nonunital+ring">current</a>

   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

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeSep 4th 2023
   • (edited Sep 4th 2023)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 5th 2023
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI 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.

   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.