[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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895(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/29(土)13:30 ID:T0GrcKp2(6/15) AAS
>>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
省17

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896(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/29(土)13:35 ID:T0GrcKp2(7/15) AAS
>>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
省23

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