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)
上下前次1-新
通常表示 512ﾊﾞｲﾄ分割ﾚｽ栞
抽出解除ﾚｽ栞

このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索歴削→次ｽﾚ栞削→次ｽﾚ過去ﾛｸﾞﾒﾆｭｰ

895: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/08/29(土) 13:30:50.73 ID:T0GrcKp2

>>894
つづき

Suppose R is a division ring. We claim that the R -module R has the following
properties: (a) R ± (0) and (b) (0) and R are the only submodules of R. By
definition, a division ring R is not zero so (a) is trivially satisfied. Suppose now
that M is a nonzero submodule of a division ring R. Then there is a nonzero x in
M. Because R is a division ring there is a y in R such that yx = 1. Because yx is in
M, it follows that 1 is in M and so r=r\ is in M for all r in R, which means that
M = R. So we see that a division ring R also satisfies (b), that is, it has the
property that (0) and R are the only submodules of R.
On the other hand, it is not difficult to see that a nonzero ring R which has the
property that (0) and R are its only submodules, is a division ring. To show this we
first show that if x is a nonzero element of R and yx = 0, then y = 0. The set M of
all y in R such that yx = 0 is a submodule of R, because it is the kernel of the
morphismof R -module R →R given by rl→ncfora!l rin R. Now M±R because 1
is not in R (remember R is not the zero ring). Therefore, M = (0) because (0) and R
are the only submodules of R. Hence, if yx = 0, then y = 0 because it is in M and
M = (0).

Next we observe that if x is a nonzero element of R, then there is a y in R
such that yx = 1. For the subset Rx is a submodule of R which is not the zero
submodule of R because it contains the nonzero element x. Hence, Rx = R be
cause (0) and R are the only submodules of R and Rx±0. This means that there is
a y in R such that yx = 1.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1595166668/895

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

上下前次1-新書関写板覧索設栞歴

ｽﾚ情報赤ﾚｽ抽出画像ﾚｽ抽出歴の未読ｽﾚ AAｻﾑﾈｲﾙ

ぬこの手ぬこTOP 0.170s*