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Now I’m looking at the article radical. This comment might be mostly for Zoran (especially for the noncommutative case), but anyone may reply obviously.
The condition of idempotence $\sigma \sigma = \sigma$ bothers me a little; I don’t think it’s a coherent notion as given (what is this equality between functors). It’s fine though if $\sigma$ is given as a subfunctor of the identity functor, which I guess is the main intention of this area, so is there any objection if I move the phrases around to make it come out like that?
What I’d really want to get to in this article is some discussion of radicals as closure operators on ideal lattices. There’s a two-sorted notion of module (a ring R together with an $R$-module $M$), and maybe somewhere in the literature, although I don’t know where, there would be a notion of radical functor $\sigma$ on $Mod$ which would send a pair $(R, M)$ to $(R, \sigma(M) \subseteq M)$. (In other words, we don’t work in just one fiber $Mod_R$ at a time, but over all fibers simultaneously.) If $I$ is a two-sided ideal of $R$, then we could consider $\sigma(R/I, R/I) = (R/I, \sigma(R/I))$ and then pull back the ideal $\sigma(R/I)$ along the quotient map $R \to R/I$ to get an ideal containing $I$. This could be considered the $\sigma$-closure of an ideal. For example, the Jacobson radical of an ideal should fit nicely in some such context (the intersection of all maximal deals containing $I$).
I am just learning now of some really nice properties of some of these radicals. For example, according to some work of Banaschewski and Harting, the Moore closure operators $k$ on ideal lattices defined by the Jacobson, Levitski, and Brown-McCoy radicals have the following properties (in addition to being finitary = preserving directed joins):
$k(I \cap J) = k(I J) = k(I) \cap k(J)$,
$k(J) = 1$ implies $J = 1$ (the top element).
These properties imply that the radical ideals (= fixed points of $k$) form a compact frame (i.e., the top element $1$ is compact), and from there one can do some nice things, for example, deduce the existence of a point, or a prime ideal. I’ve been writing this up at prime ideal theorem.
I found this post rather late in the day. I will be happy to help to improve that page and to communicate to you about the closure operation aspect as I did recently revisit this topic (much after the page is written). For the discussion, insights in the following article may be useful:
Thanks, Zoran! Unfortunately I can see only two pages of the article for free.
OK, I will send you the file
I looked again at radical and moved some phrases around to make some things clearer. However, nowhere on the page is the general notion of radical functor defined, and I’m not sure what Zoran intended. Would be nice to clear this up.
We do have something at Jacobson radical, so I added cross-links. But of course currently these entries do not talk to each other.
Thanks, Urs. I’m working on radical now, and will report back later (I think I know what Zoran was saying).
I’ve attempted to clarify to myself the literature mess that Zoran was pointing to, but without having my hands on much of the literature myself. I’ll have to look at it further (and I hope Zoran will see this and check what I’ve done), but I’m done for the time being.
It looks OK (we can not rectify the chaos in the literature far from recording the range of possibilities, it seems). I changed the section title “Properties” into “Radical functors” as it seems more appropriate. I added the missing Goldman’s reference under the literature section.
Thanks, Zoran! I want to do more with this article (especially: drawing the connection between notions of radical in the definition section and radical functors, and giving more examples).
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