[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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898(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/29(土)13:37 ID:T0GrcKp2(9/15) AAS
>>897
つづき

Hence, R is a division ring because we have already seen that a ring R is a division
ring if and only if the R-module R is a simple R-module. Thus, our problem of
showing that a nonzero ring R is a division ring if every R-module is free is solved
once we establish that every nonzero ring has a simple module. To this end it is
convenient to have the following.
省22

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899: 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/29(土)13:38 ID:T0GrcKp2(10/15) AAS
>>898
つづき

P221
Because F is an inductive set, it must have a maximal element M by Zorn's
lemma. We leave it to the reader to verify that M is a maximal submodule of R.
Because M obviously contains M', the first part of the proposition is proven.
In the light of this result, to see that R contains at least one maximal submodule,
省11

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