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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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

894: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/08/29(土) 13:29:38.29 ID:T0GrcKp2

>>893
つづき

Basic Properties 8.1
Let M be an R-module.

(b) Every linearly independent subset S of M is contained in a maximal linearly
independent subset of M.
(c) M has a maximal linearly independent subset.
(d) If M is a free R-module, then every basis for M is a maximal linearly indepen
dent subset of M.
PROOF: 略

P217
Proposition 8.2
Let D be a division ring. Then the following statements are equivalent for a subset
B of a D-module M:
(a) B is a basis for M.
(b) B is a maximal linearly independent subset of M. Since every module has a
maximal linearly independent subset, every module over a division ring D has
a basis and is therefore a free D-module.
PROOF: Because the reader has already shown that every basis of a module is
a maximal linearly independent subset of the module, we only have to show that
(b) implies (a).
略
P218
This finishes the proof that a maximal linearly independent subset B of a
D-module M is a basis for M because it generates M.

Having established that all modules over division rings are free, we will have
a complete description of all nonzero rings R with the property that all R -modules
are free if we show that any nonzero ring with this property must be a division
ring. Because we are trying to describe when a ring is a division ring in terms of its
module theory, it is reasonable to expect that a module-theoretic description of
when a ring is a division ring would be helpful. We do this now in terms of the
properties of the R-module R.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1595166668/894

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