[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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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
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897(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/29(土)13:37 ID:T0GrcKp2(8/15) AAS
>>896
つづき

Basic Properties 8.4
Let R be an arbitrary ring and M a nonzero R-module. The following conditions
are equivalent:
(a) M is generated by each nonzero element in M.
(b) For every R-module X, every morphism f:X→M is either zero or an
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