[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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894(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/29(土)13:29 ID:T0GrcKp2(5/15) AAS
>>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.
省26

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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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