ガロア第一論文と乗数イデアル他関連資料スレ3

[過去ﾛｸﾞ] ガロア第一論文と乗数イデアル他関連資料スレ3 (1002ﾚｽ)
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277: １３２人目の素数さん [] 2023/04/21(金) 06:48:35.37 ID:vIwU6BoW

>>247 追加

＜ああ おサルの勘違い2＞
用語"cancellable"について
前スレ https://rio2016.5ch.net/test/read.cgi/math/1680684665/146-147 2023/04/13 より
（おサル）
>　上記の英文の正しい訳ｈ以下の通りです
>「左零因子でない環の元は、左正規もしくは左キャンセル可能と呼ばれる」
>　つまり、zero divisorの否定だけです
>　それをregular、または同じことですが、cancellable　と呼んでいるのです
>　したがって、cancellableについての以下の憶測は完全な誤りです
>>”cancellable”とは、乗法の逆元を持つことで、”cancel”可能と解釈したけど

（私）
en.wikipediaの記事だけに頼ると、嵌まるよｗ

regular "cancellable" ring zero divisor
での検索で下記文献ヒット

1）”cancellable”の定義見つけたよ（下記 Henri Bourles）
（そもそも、>>143のen.wikipediaには、文献[3]Nicolas Bourbaki (1998). Algebra I. Springer Science+Business Media. p. 15.とあるよね？
　それをチェックしないで短絡はダメじゃんｗ）
2）cancellable：”xy = xz ⇒ y = z”とあるよ。これ大事だな
3）それから、用語Regularの説明は、下記Darij Grinbergの「Regular elements of a ring, monic polynomials and “lcm-coprimality”」見てね
4）要するに、ｎ次正方行列から、regularを取り除くとzero divisorに、逆にzero divisorを取り除くとregularに
　この関係がキモですよ

https://www.sciencedirect.com/topics/mathematics/zero-divisor
Elementary Algebraic Structures
Henri Bourles, in Fundamentals of Advanced Mathematics, 2017

2.1.1 Monoids and divisibility
(II) Divisibility. In the rest of this subsection, monoids are written multiplicatively and have zeros.
An element x ∈ M× is said to be left-cancellable (resp. right-cancellable) if xy = xz ⇒ y = z (resp. yx = zx ⇒ y = z) and cancellable if it is both left- and right-cancellable. A monoid M with the property, that every element of M× is cancellable, is said to be a cancellation monoid.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1680684665/277

278: １３２人目の素数さん [] 2023/04/21(金) 06:49:05.65 ID:vIwU6BoW

>>277
つづき

https://www.cip.ifi.lmu.de/~grinberg/algebra/regpol.pdf
Regular elements of a ring, monic polynomials and “lcm-coprimality”
Darij Grinberg May 22, 2021

P5
2. Regular elements (a.k.a. non-zero-divisors)
2.1. Definition
We begin with a basic notation:
Definition 2.1. Let A be a commutative ring. Let a ∈ A.
The element a of A is said to be regular if and only if every x ∈ A satisfying ax = 0 satisfies x = 0.
Instead of saying that a is regular, one can also say that “a is cancellable”, or that “a is a non-zero-divisor”.

This notion of “regular” elements has nothing to do with various other notions of “regularity” in commutative algebra (for example, it is completely unrelated to the notion of a “von Neumann regular element” of a ring).
It might sound like a bad idea to employ a word like “regular” that has already seen so much different uses; however, we are not really adding a new conflicting meaning for this word, because the word is already being used in this meaning by various authors (among them, the authors of [LLPT95]), and because our use of “regular” is closely related to the standard notion of a “regular sequence” in a commutative ring 4.
Many authors (for example, Knapp in [Knapp2016]) define a zero divisor in a commutative ring A to be a nonzero element of A that is not regular.5 Thus, at
least in classical logic, regular elements are the same as elements that are not zero divisors (with the possible exception of 0). I find the notion of a “zero divisor”less natural than that of a regular element (it is the regular elements, not the zero divisors, that usually exhibit the nicer behavior), and it is much less suitable for constructive logic (as it muddies the waters with an unnecessary negation), but it appears to be more popular for traditional reasons.
(引用終り)
以上

結論：用語"cancellable"の意味が理解できないおサルさんでしたとさ

http://rio2016.5ch.net/test/read.cgi/math/1680684665/278

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