Thomas Klimpel
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 Dec 8 awarded Caucus Nov 29 comment Which associative and commutative operations are defined for any commutative ring? @ThomasAndrews Yes, in case of addition the corresponding condition is that $b$ has to be in the range of $x\to ax$. What I like less is that $a$ and $b$ are currently not true parameters, but are restricted to the free commutative ring without generators, i.e. $\mathbb Z$. But I'm not sure how to extend Martin Brandenburg's "natural" argument to the case with parameters. Nov 29 comment Which associative and commutative operations are defined for any commutative ring? @ThomasAndrews It certainly won't be in the range of $\phi_{ab}$ if $b(b-1)/a$ is not defined, i.e. when $b(b-1)$ is not in the range of $x\to ax$. If on the other hand $b(b-1)$ is in the range of $x\to ax$, then the laws of commutative rings are sufficient to show that also the other expression will work. Nov 29 comment How to show distributivity in a ring, and what is wrong with my algebra? See also the appendix of math.stackexchange.com/questions/1001186/… Nov 27 awarded Nice Answer Nov 12 revised Which associative and commutative operations are defined for any commutative ring? Added an appendix with a more intuitive explanation/description of the found operations Nov 6 comment Graph-Minor Theorem for Directed Graphs? @Denis My guess would be that the other answer was the first answer by a new user, and got its votes through the corresponding review queue. In the review queue, you can only see the answer and the question, but not the other answers. The main claim I make in this answer is that the definition "A directed graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges." is equivalent to the definition given in the question. If you want to challenge this claim, or think I should it explain it in more detail, please let me know. Nov 3 comment Example of a commutative ring which is not a subring of a commutative ring where every non-invertible element is a zero-divisor Thanks for the link. So instead of "every non-invertible element is a zero-divisor", I could say "every regular element is a unit". Sweet, short and positive... Nov 3 accepted Example of a commutative ring which is not a subring of a commutative ring where every non-invertible element is a zero-divisor Nov 3 asked Example of a commutative ring which is not a subring of a commutative ring where every non-invertible element is a zero-divisor Nov 2 revised Which associative and commutative operations are defined for any commutative ring? Now M.B. explained to me why the "if x is unit then f(x) else g(x)" construct is not natural... Nov 2 accepted Which associative and commutative operations are defined for any commutative ring? Nov 2 comment Which associative and commutative operations are defined for any commutative ring? Wow, you really taught me some category theory today. The fact that ring homomorphisms don't preserve non-units is so typical, yet I still fall into the "law of excluded middle" trap. And the lesson about natural transformations and representation of the forgetful functor was most welcome, since "theoretically I mastered that material some time ago", but this "practical application" was still challenging for me. It took me more than an hour and some repetition of material which I "believed to master", before I got the warm fuzzy feeling that I understood what you told me... Nov 1 comment Which associative and commutative operations are defined for any commutative ring? Maybe the "if $x\in R^*$ then $f(x)$ else $g(x)$" can't be interpreted as a natural transformation, but before we can even ask whether an operation is a natural transformation, we have to fix the source and target functor. This was the crucial point why it took me so long to come to terms with adjoint functors: I kept thinking about natural transformations without taking care that the source and target functor are as crucial as the family of morphisms itself. Your diagram looks as if the target functor is an arbitrary $\mathcal F$, and the source is $\mathcal F \times \mathcal F$. Correct? Nov 1 comment Which associative and commutative operations are defined for any commutative ring? It should be possible to define operations in rings with the help of an "if $x\in R^*$ then $f(x)$ else $g(x)$" construct, where $R^*$ is the group of units of $R$. You argued that this construct is not natural in the sense of category theory, but I fail to see why it should not be natural, as long as $f(x)$ and $g(x)$ are natural. The property of being an element of the group of units seems to be invariant under (unitary) ring homomorphism to me. Nov 1 revised Which associative and commutative operations are defined for any commutative ring? edited body Nov 1 revised Which associative and commutative operations are defined for any commutative ring? fixed sign mistake and updated question with information from the comments Nov 1 comment Which associative and commutative operations are defined for any commutative ring? @MartinBrandenburg I think I already found a counterexample to my question. Take the unary operation $u(x)$ which maps any element of the group of units to $1$, and all other elements to $0$. Then $u(xy)$ is an associative and commutative binary operation not covered by $x_{abc}y$. By "operation", I mean that only the ring structure is used, and not an order or field structure, which wouldn't be available in a general ring. But I don't put any special requirements on $a, b, c$, like being simultaneously defined for all rings. (I downloaded you reference and will read it now.) Nov 1 asked Which associative and commutative operations are defined for any commutative ring? Oct 31 answered Can the b-adic representation of rational numbers (by quote notation) be extended to non-terminating expansions?