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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896: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/08/29(土) 13:35:40.52 ID:T0GrcKp2

>>895
つづき

We now show that these two observations imply that a nonzero ring R is a
division ring if (0) and R are the only submodules of R. To do this we must show
that if x is a nonzero element of R, then there is a y in R such that yx = 1 = xy. By
what we have just shown we know that if x is a nonzero element of R, then there
is a y in R such that yx = 1. Multiplying both sides of this equation by y on the
right, we obtain yxy = y or, equivalently, y(xy - 1) = 0. The fact that R is not the
zero ring means that 1≠0. Because yx = 1, it follows that y ≠0. But this, combined
with the fact that y(xy-l) = 0, implies that xy-l=0. For if xy-l≠0, then by
previous observation we would have y(xy- 1)≠0 because both y and xy- 1 are
different from zero. Hence, xy = 1 which gives our desired result that yx = 1 = xy.
Thus, we have shown that a nonzero ring R is a division ring if (0) and R are the
only submodules of R.
We summarize our discussion up to this point in the following.

P219
Proposition 8.3
A ring R is a division ring if and only if the R-module R is a nonzero module
satisfying the condition that (0) and R are the only submodules of R.
This result suggests that for an arbitrary ring R the nonzero R -modules M
with the property that (0) and M are the only submodules of M, might be worth
considering. In fact they play an important role in all of ring theory and for this
reason are given a special name.

Definition
Let R be an arbitrary ring. An R -module M is called a simple R-module if M=f=(0)
and (0) and M are the only submodules of M.
In this terminology our previous result becomes: A ring R is a division ring if
and only if the R-module R is a simple R-module. We leave it to the reader to
verify the following characterization of simple R-modules.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1595166668/896

897: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/08/29(土) 13:37:06.72 ID:T0GrcKp2

>>896
つづき

Basic Properties 8.4
Let R be an arbitrary ring and M a nonzero R-module. The following conditions
are equivalent:
(a) M is generated by each nonzero element in M.
(b) For every R-module X, every morphism f:X→M is either zero or an
epimorphism.
(c) For every R-module X, every morphism f:M→X is either zero or a
monomorphism.
As an immediate consequence of these basic properties, we have the
following.

Corollary 8.5
Let M be a simple R-module. Then every endomorphism of M is either zero or an
automorphism. Hence, EndR( M ), the ring of endomorphisms of M, is a division
ring.
The main point to establish about simple modules in connection with our
problem of showing that a nonzero ring R is a division ring if every R-module is
free is that every nonzero ring R has at least one simple R-module.
Suppose we know that our nonzero ring R, which has the property that every
R-module is free, also has a simple R-module M. Then the simple R-module M
must have a basis B since M is a free R-module. Because M ≠ (0), we know that
B is not empty. We now show that B consists of exactly one element. Let b be an
element of B. Then by one of our characterizations of simple modules (Basic
Property 8.4), we know that the element b generates M since b ± 0. By Basic
Property 7.6, it follows that {b} = B. Hence, B consists of a single element.
But we have already shown that a free module over a ring R has a basis
consisting of one element if and only if it is isomorphic to R. Hence, the simple
R-module M is isomorphic to R which means that R is a simple R-module.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1595166668/897

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