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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 24th 2016
   • (edited Jan 24th 2016)
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   Author: Todd_Trimble
   Format: MarkdownItexIn the article [[ideal]], I do not think the definition of prime ideal for rings or rigs is stated accurately. Certainly the definition is appropriate for *commutative* rings or rigs, but for noncommutative rings at least, the usual definition is different: an ideal $P$ is prime if for any ideals $A, B$, if $A B \subseteq P$, then $A \subseteq P$ or $B \subseteq P$. Under this definition, we have that the zero ideal in a matrix ring over a field is a prime ideal (vacuously, since matrix rings are simple rings) -- but it is not prime under the nLab definition. I'm not sure what the definition ought to be in the case of noncommutative rigs. Maybe it's the same. (I generally wonder how much we should be focused on ideals in rigs, or similarly on kernels of maps between commutative monoids, when it seems to me that congruences or kernel pairs are the more salient structural notion for these cases.)

   In the article ideal, I do not think the definition of prime ideal for rings or rigs is stated accurately. Certainly the definition is appropriate for commutative rings or rigs, but for noncommutative rings at least, the usual definition is different: an ideal $P$ is prime if for any ideals $A, B$, if $A B \subseteq P$, then $A \subseteq P$ or $B \subseteq P$. Under this definition, we have that the zero ideal in a matrix ring over a field is a prime ideal (vacuously, since matrix rings are simple rings) – but it is not prime under the nLab definition.

   I’m not sure what the definition ought to be in the case of noncommutative rigs. Maybe it’s the same. (I generally wonder how much we should be focused on ideals in rigs, or similarly on kernels of maps between commutative monoids, when it seems to me that congruences or kernel pairs are the more salient structural notion for these cases.)

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJan 24th 2016
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   Author: Mike Shulman
   Format: MarkdownItexInteresting! Why is that the usual definition? I don't think I would have expected the zero ideal to be prime in a ring (commutative or otherwise) that has zero divisors.

   Interesting! Why is that the usual definition? I don’t think I would have expected the zero ideal to be prime in a ring (commutative or otherwise) that has zero divisors.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 24th 2016
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   Author: Todd_Trimble
   Format: MarkdownItexThat's a very good question, but I'm afraid I don't have a good answer. My knowledge of noncommutative algebra is embarrassingly close to zero. I'm leafing through Google books by [Lam](https://books.google.com/books?id=2PwGCAAAQBAJ&pg=PA165&lpg=PA165&dq=prime+ideals+noncommutative+rings&source=bl&ots=Qikg9aD22p&sig=_eb1BjB2BJvLBsTZW88Nh0Cgp4g&hl=en&sa=X&ved=0ahUKEwi5oNOZvcPKAhWCdj4KHX4GBj8Q6AEIZzAJ#v=onepage&q=prime%20ideals%20noncommutative%20rings&f=false) and [Goodearl and Warfield](https://books.google.com/books?id=vXh1MtvQr_8C&pg=PA47&lpg=PA47&dq=prime+ideals+noncommutative+rings&source=bl&ots=d33dJwwtwU&sig=Q7NOLP3lyA27C5kMIbHHJR4Gqbg&hl=en&sa=X&ved=0ahUKEwi5oNOZvcPKAhWCdj4KHX4GBj8Q6AEIYzAI#v=onepage&q=prime%20ideals%20noncommutative%20rings&f=false) to try to learn more. Zoran could tell us a great deal, I'd bet. My rough impression is that a large bulk of noncommutative ring theory is developed by studying their module categories, and that the conventional definition of prime ideal has proven its worth by being the more useful one for describing module structure. Maybe those category theorists who go one step further from noncommutative rings to algebroids have also developed ideas about ideal theory.

   That’s a very good question, but I’m afraid I don’t have a good answer. My knowledge of noncommutative algebra is embarrassingly close to zero. I’m leafing through Google books by Lam and Goodearl and Warfield to try to learn more.

   Zoran could tell us a great deal, I’d bet. My rough impression is that a large bulk of noncommutative ring theory is developed by studying their module categories, and that the conventional definition of prime ideal has proven its worth by being the more useful one for describing module structure. Maybe those category theorists who go one step further from noncommutative rings to algebroids have also developed ideas about ideal theory.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 31st 2016
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   Author: Todd_Trimble
   Format: MarkdownItexSo getting back to some of these issues, I think maybe the way to handle the notion of "prime ideal" is to (1) note the the lattice of ideals for many of the situations of genuine interest (rings, rigs, distributive lattices) forms a [[quantale]], and (2) note that there is a general notion of prime element in a quantale ($p$ such that $x y \leq p$ implies $x \leq p$ or $y \leq p$) which specializes to the 'correct' notion of prime ideal in each of these cases. What is additionally nice is that there seems to be a general result that a nontrivial compact quantale admits a prime element. This can be regarded as a fairly general [[prime ideal theorem]] which covers all the cases set out in the introduction of that article. Finally, in partial reply to Mike's question in #2, I'll just note that maximal (two-sided) ideals in rings are prime if we use this 'correct' notion, but not if we use the original definition given in the nLab. (More commentary on this question may have to wait. Somewhat annoyingly, perhaps, is that prime ideals are not stable under inverse image.)

   So getting back to some of these issues, I think maybe the way to handle the notion of “prime ideal” is to (1) note the the lattice of ideals for many of the situations of genuine interest (rings, rigs, distributive lattices) forms a quantale, and (2) note that there is a general notion of prime element in a quantale ($p$ such that $x y \leq p$ implies $x \leq p$ or $y \leq p$) which specializes to the ’correct’ notion of prime ideal in each of these cases.

   What is additionally nice is that there seems to be a general result that a nontrivial compact quantale admits a prime element. This can be regarded as a fairly general prime ideal theorem which covers all the cases set out in the introduction of that article.

   Finally, in partial reply to Mike’s question in #2, I’ll just note that maximal (two-sided) ideals in rings are prime if we use this ’correct’ notion, but not if we use the original definition given in the nLab. (More commentary on this question may have to wait. Somewhat annoyingly, perhaps, is that prime ideals are not stable under inverse image.)