純粋・応用数学（含むガロア理論）3

[過去ﾛｸﾞ] 純粋・応用数学（含むガロア理論）3 (1002ﾚｽ)
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891: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/08/29(土) 13:26:12.38 ID:T0GrcKp2

>>724
(引用開始)
https://ja.wikipedia.org/wiki/%E6%96%9C%E4%BD%93_(%E6%95%B0%E5%AD%A6)#CITEREFAuslanderBuchsbaum2004
斜体 (数学)
斜体であるという性質は加群の圏の性質から特徴づけることもできる。環 R が斜体である必要十分条件はすべての左 R 加群が自由加群であることである[5]。
出典
5.^ Auslander & Buchsbaum 2004, p. 221, Theorem 6.8.8.
参考文献
・Auslander, Maurice; Buchsbaum, David (2014). Groups, Rings, Modules. Dover. ISBN 978-0-486-49082-3. MR0366959. Zbl 0325.13001

読んだ
なかなか面白かったな
以下要点抜粋（興味のある方は本文をどぞ）

>https://upload.wikimedia.org/wikipedia/commons/a/a0/Groups_Rings_Modules_Auslander_Hathitrust.pdf
Groups, Rings, Modules. Auslander, Maurice; Buchsbaum, David (1974).
(抜粋)
P57
Definition
Let R be a ring. By an R -module structure on an abelian group M we mean a map
RxM→M which we denote by (r, m)→rm satisfying:
(a) (r1 + r2)m = r1m + r2m.
(b) r(m1 + m2) = rm1 + rm2.
(c) (r1r2)(m) = r1(r2m).
(d) 1m = m.

An abelian group together with an R-module structure is called an It-module.
We shall return later on to this general notion of a module. In fact, most of
this book will be devoted to a detailed study of rings and modules.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1595166668/891

892: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/08/29(土) 13:27:18.99 ID:T0GrcKp2

>>891
つづき

P210
7. FREE R-MODULES
Undoubtedly, the modules that are most familiar to the reader are vector spaces
over a field. Probably the most distinctive feature of the theory of vector spaces is
that every vector space has a basis. After generalizing the notion of a basis from
vector spaces over fields to modules over arbitrary rings, we introduce the notion
of a free module over an arbitrary ring. Namely, an R -module is a free R -module
if and only if it has a basis.

Definition
Let M be an R-module. A subset S of M is said to be a linearly independent subset
of M if each finite subset of distinct elements s1, . . . , sn in S has the property that
given r1, . . . , rn in R such that Σi=1〜n risi =0, then each ri = 0 for i = 1, . . . , n.

Before giving examples of linearly independent subsets of modules, it is
convenient to have the following easily verified properties.

Basic Properties 7.1
Let M be an R-module.
(a) The empty set is a linearly independent subset of M.
(b) A subset S of M consisting of a single element m is a linearly independent
subset of M if and only if for any r in R we have rm = 0 implies r = 0. Hence,
the subset {m} is linearly independent if and only if the morphism of R-modules
R→M given by r→rm is a monomorphism.
(c) If S is a linearly independent subset of M, then every subset of S is also a
linearly independent subset of M.
(d) For a subset S of M, the following statements are equivalent:
(i)S is a linearly independent subset of M.
(ii) Each finite subset of S is a
linearly independent subset of M.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1595166668/892

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