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643(3): １３２人目の素数さん [] 2023/03/21(火)17:43 ID:8s9PZXQ2(15/20)
>>642
つづき
https://en.wikipedia.org/wiki/Commutative_algebra
Commutative algebra
This article is about a branch of algebra. For algebras that are commutative, see Commutative algebra (structure).
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers Z ; and p-adic integers.
Commutative algebra is the main technical tool in the local study of schemes.
The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras.
Overview
Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry.
In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings.
Connections with algebraic geometry
Commutative algebra (in the form of polynomial rings and their quotients, used in the definition of algebraic varieties) has always been a part of algebraic geometry. However, in the late 1950s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a scheme. Their local objects are affine schemes or prime spectra, which are locally ringed spaces, which form a category that is antiequivalent (dual) to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field k, and the category of finitely generated reduced k-algebras.
つづく
644(2): １３２人目の素数さん [] 2023/03/21(火)17:44 ID:8s9PZXQ2(16/20)
>>643
つづき
The gluing is along the Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes.
https://ja.wikipedia.org/wiki/%E5%8F%AF%E6%8F%9B%E7%92%B0%E8%AB%96
可換環論（英語：commutative algebra、commutative ring theory）は、その乗法が可換であるような環（これを可換環という）に関する理論の体系のこと、およびその研究を行う数学の一分野のことである。
https://en.wikipedia.org/wiki/Associative_algebra
Associative algebra
This article is about a particular kind of algebra over a commutative ring. For other uses of the term "algebra", see Algebra (disambiguation).
In mathematics, an associative algebra A is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field K. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication.
https://ja.wikipedia.org/wiki/%E7%B5%90%E5%90%88%E5%A4%9A%E5%85%83%E7%92%B0
結合多元環
(引用終り)
以上
645: １３２人目の素数さん [sage] 2023/03/21(火)17:46 ID:030eOzSs(11/16)
>>641-644
アホ1
全く理解できないネタで粋がる
正真正銘の●違い
647: １３２人目の素数さん [] 2023/03/21(火)17:49 ID:8s9PZXQ2(17/20)
Commutative algebra>>643に、Associative algebra,結合多元環>>644
か
随分新しい数学用語が増えていますね
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