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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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

893: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/08/29(土) 13:28:29.84 ID:T0GrcKp2

>>892
つづき

(iii) If {rs}s∈S is an almost zero family of elements of R such that Σs∈S rss =0,
then rs =0 for each s in S.
(iv) If {rs}s∈S and {r's}s∈S are two almost zero families of elements of R such
that Σs∈S rss = Σs∈S r'ss, then rs=r's for all s in S.

We now give some examples to illustrate various types of linearly indepen
dent subsets of modules.

P216
8. CHARACTERIZATION OF DIVISION RINGS

We now turn our attention to describing those rings R with the property that every
R-module is a free R-module. The reader is already familiar with the fact that
fields have this property. We now show that division rings, which are the natural
generalization of the notation of a field to arbitrary, not necessarily commutative,
rings, also have the same property.

Definition
A ring R is called a division ring if it is not the zero ring and every nonzero element
in R is a unit in R.

Obviously, a commutative ring is a division ring if and only if it is a field. So
fields are special cases of division rings.
In order to show that every module over a division ring has a basis, it is
convenient to have the notion of a maximal linearly independent subset of a
module over an arbitrary ring R.

Definition
A subset 5 of an R- module M is said to be a maximal linearly independent subset
of M if S is linearly independent and S is not contained in any larger linearly
independent subset of M.

The main result about maximal linearly independent subsets of a module M is
that every linearly independent subset of M is contained in a maximal such subset
of M. The proof of this fact is the burden of the following.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1595166668/893

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