[過去ﾛｸﾞ] 純粋・応用数学（含むガロア理論）3 (1002ﾚｽ)

純粋・応用数学（含むガロア理論）3 http://rio2016.5ch.net/test/read.cgi/math/1595166668/

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635: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2020/08/23(日) 16:29:40.05 ID:ehdjUjVy >>630 追加 これも、メモ https://en.wikipedia.org/wiki/Divisibility_(ring_theory) Divisibility (ring theory) In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. With the development of abstract rings, of which the integers are the archetype, the original notion of divisor found a natural extension. Divisibility is a useful concept for the analysis of the structure of commutative rings because of its relationship with the ideal structure of such rings. Definition Let R be a ring,[1] and let a and b be elements of R. If there exists an element x in R with ax = b, one says that a is a left divisor of b in R and that b is a right multiple of a.[2] Similarly, if there exists an element y in R with ya = b, one says that a is a right divisor of b and that b is a left multiple of a. One says that a is a two-sided divisor of b if it is both a left divisor and a right divisor of b; in this case, it is not necessarily true that (using the previous notation) x=y, only that both some x and some y which each individually satisfy the previous equations in R exist in R. When R is commutative, a left divisor, a right divisor and a two-sided divisor coincide, so in this context one says that a is a divisor of b, or that b is a multiple of a, and one writes a ｜ b. Elements a and b of an integral domain are associates if both a ｜ b and b ｜ a. The associate relationship is an equivalence relation on R, and hence divides R into disjoint equivalence classes. Notes: These definitions make sense in any magma R, but they are used primarily when this magma is the multiplicative monoid of a ring. つづく http://rio2016.5ch.net/test/read.cgi/math/1595166668/635

636: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2020/08/23(日) 16:30:07.34 ID:ehdjUjVy >>635 つづき Zero as a divisor, and zero divisors ・Some authors require a to be nonzero in the definition of divisor, but this causes some of the properties above to fail. ・If one interprets the definition of divisor literally, every a is a divisor of 0, since one can take x = 0. Because of this, it is traditional to abuse terminology by making an exception for zero divisors: one calls an element a in a commutative ring a zero divisor if there exists a nonzero x such that ax = 0.[3] (引用終り) 以上 http://rio2016.5ch.net/test/read.cgi/math/1595166668/636

637: １３２人目の素数さん [sage] 2020/08/23(日) 17:02:18.03 ID:7NMituVg >>635-636 貴様、なにがしたいの？ http://rio2016.5ch.net/test/read.cgi/math/1595166668/637

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