I think you don't have Weyl algebras in $2n$ variables, but don't know if they count. Besides, in positive characteristic there are differing definitions of Weyl algebras
Neat! Well it's nice to know you better. I'm happy to talk about rings and modules of course, but I'm also interested in geometry and mathematical physics, and also Python programming, which I do professionally now.
The original paper of Kaplansky is Rings with a polynomial identity (1948). The theorem is in all serious books about PI rings. For example, in Rowen's Polynomial identities in ring theory it is Theorem 1.5.16
Kaplansky was a very interesting person. One of those people who you discover thought about a lot of things that occurred to you (and a bunch more things you didn't)
I think the database needs a nonsimple positive characteristic weyl algebra
@JoseBrox I have only tinkered around a little with Sage. A while ago I was using it to search for small examples to a ring-theoretic question from my dissertation
@JoseBrox I'd probably be happy for excuses to use it more
You see, if char(K)=p, then [y,x^p]=px^(p-1)=0, so x^p commutes with y, but also with x, hence with the whole algebra A. Then the ideal generated by x^p is just x^pA, and x is not in there due to degree, that can be defined in this context also
I don't think you will have them disguised. Weyl algebras are differential polynomial rings, but I'm not sure if they are still in char p or not. There is yet another definition, but I don't recall it now
"They are differential polynomial rings, but i'm not sure if they are in char p" . Hm, this treats "differential polynomial ring" as a property but I was thinking about it as a construction. What am I missing?
I mean that Weyl algebras as you have them defined (well, you get x1,...,xn and y1,...,yn where yi is the derivative of xi) are isomorphic to differential polynomial rings (with the usual derivative as the twisting derivation), but I'm not sure if this is true anymore in char p. In a quick look, they seem to be the same, but...
I think I would maintain the same page, and distinguish the properties by the characteristic, as in Simple: yes (char 0) / no (char p)
I'll leave for now. My connection problems are really killing me... but you can write anything in here and I will read it later, or you can write to me at [email protected] if you prefer
@JoseBrox I can explain that. In order to be able to list its cardinality, I made the field countable. Typically I make them as small as possible, unless something larger is needed. I remember specifically one thing that required an uncountable field in its construction to differentiate it nontrivially from the same construction with a countable field.
@JoseBrox Structurally that would be difficult :) especially if it turns out they have more nontrivial differences
@Zacky Aha, thanks for indirectly complimenting my site's name. I wanted to name it "All Rings Considered" but it turns out that's already taken (apparently it's a pro-wrestling blogger)
@JoseBrox I think I see what you're getting at: I might have confused two things in my description. On one hand I called them differential operators, but in the description it's that quotient. Would it be most consistent to just call it the first Weyl Algebra and stick to the quotient?