[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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737(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/27(木)11:49 ID:NVBIr97s(1/12) AAS
>>736 追加
外部ﾘﾝｸ:planetmath.org
rings whose every module is free Author joking (16130)
Last modified on 2013-03-22

この前半の証明も良いね〜

”Recall that if R is a (nontrivial) ring and M is a R-module, then (nonempty) subset S⊆M is called linearly independent if for any m1,…,mn∈M
and any r1,?,rn∈R the equality
省8

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738(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/27(木)11:50 ID:NVBIr97s(2/12) AAS
>>737
つづき

Proof. ,,⇒” First assume that R is a divison ring.
Then obviously R has only two (left) ideals, namely 0 and R
(because every nontrivial ideal contains invertible element and thus it contains 1, so it contains every element of R).
Let M be a R-module and m∈M such that m≠0.
Then we have homomorphism of R-modules f:R→M such that f(r)=r・m.
省18

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