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2010年1月30日土曜日
数学学習の記録 77 加群としてのR-準同型写像(線形写像)について。
GoogleドキュメントのTeXによる数式入力の練習。
Rを環(Ring),
M,M'をR上の加群とする。
M'についても同様。
MからM'への写像fがR-準同型写像(R-線形写像)。
ベクトル空間と同様に、fの像f(M)はM'の部分加群、
fの核(Kernel)はMの部分加群、
となり、f(M)は商加群M/Ker fとR-同型となる。
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![(1)\forall a,b,c\in R[(a+b)+c=a+(b+c)]\\
(2)\exists 0\in R\forall a\in R[a+0=0+a=a]\\
(3)\forall a\in R\exists -a\in R[a+(-a)=(-a)+a=0]\\
(4)\forall a,b\in R[a+b=b+a]](https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=(1)%5Cforall%20a%2Cb%2Cc%5Cin%20R%5B(a%2Bb)%2Bc%3Da%2B(b%2Bc)%5D%5C%5C%0A(2)%5Cexists%200%5Cin%20R%5Cforall%20a%5Cin%20R%5Ba%2B0%3D0%2Ba%3Da%5D%5C%5C%0A(3)%5Cforall%20a%5Cin%20R%5Cexists%20-a%5Cin%20R%5Ba%2B(-a)%3D(-a)%2Ba%3D0%5D%5C%5C%0A(4)%5Cforall%20a%2Cb%5Cin%20R%5Ba%2Bb%3Db%2Ba%5D)
![(5)\forall a,b,c\in R[(ab)c=a(bc)]\\
(6)\forall a,b,c\in R\\
[a(b+c)=ab+ac\wedge (a+b)c=ac+bc]\\
(7)\exists 1\forall a\in R[1a=a1=a]](https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=(5)%5Cforall%20a%2Cb%2Cc%5Cin%20R%5B(ab)c%3Da(bc)%5D%5C%5C%0A(6)%5Cforall%20a%2Cb%2Cc%5Cin%20R%5C%5C%0A%5Ba(b%2Bc)%3Dab%2Bac%5Cwedge%20(a%2Bb)c%3Dac%2Bbc%5D%5C%5C%0A(7)%5Cexists%201%5Cforall%20a%5Cin%20R%5B1a%3Da1%3Da%5D)
![(1)\forall x,y,z\in M[(x+y)+z=x+(y+z)]\\
(2)\exists 0\in M\forall x\in M[x+0=0+x=x]\\
(3)\forall x\in M\exists -s\in M[x+(-x)=(-x)+x=0]\\
(4)\forall x,y\in M[x+y=y+z]](https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=(1)%5Cforall%20x%2Cy%2Cz%5Cin%20M%5B(x%2By)%2Bz%3Dx%2B(y%2Bz)%5D%5C%5C%0A(2)%5Cexists%200%5Cin%20M%5Cforall%20x%5Cin%20M%5Bx%2B0%3D0%2Bx%3Dx%5D%5C%5C%0A(3)%5Cforall%20x%5Cin%20M%5Cexists%20-s%5Cin%20M%5Bx%2B(-x)%3D(-x)%2Bx%3D0%5D%5C%5C%0A(4)%5Cforall%20x%2Cy%5Cin%20M%5Bx%2By%3Dy%2Bz%5D)
![(5)\forall a\in R\forall x,y\in M[a(x+y)=ax+ay]\\
(6)\forall a,b\in R\forall x\in M[(a+b)x=ax+bx]\\
(7)\forall a,b\in R\forall x\in M[(ab)x=a(bx)]\\
(8)\forall x\in M[1x=x]](https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=(5)%5Cforall%20a%5Cin%20R%5Cforall%20x%2Cy%5Cin%20M%5Ba(x%2By)%3Dax%2Bay%5D%5C%5C%0A(6)%5Cforall%20a%2Cb%5Cin%20R%5Cforall%20x%5Cin%20M%5B(a%2Bb)x%3Dax%2Bbx%5D%5C%5C%0A(7)%5Cforall%20a%2Cb%5Cin%20R%5Cforall%20x%5Cin%20M%5B(ab)x%3Da(bx)%5D%5C%5C%0A(8)%5Cforall%20x%5Cin%20M%5B1x%3Dx%5D)
![\forall a\in R\forall x,y\in M\\
[f(x+y)=f(x)+f(y)]\wedge f(ax)=af(x)]](https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cforall%20a%5Cin%20R%5Cforall%20x%2Cy%5Cin%20M%5C%5C%0A%5Bf(x%2By)%3Df(x)%2Bf(y)%5D%5Cwedge%20f(ax)%3Daf(x)%5D)
![(1)0\in f(M)\\
(2)\forall x,y\in f(M)[x+y\in f(M)]\\
(3)\forall x\in f(M)[-x\in f(M)\\
\forall a\in R\forall x\in f(M)[ax\in f(M)]](https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=(1)0%5Cin%20f(M)%5C%5C%0A(2)%5Cforall%20x%2Cy%5Cin%20f(M)%5Bx%2By%5Cin%20f(M)%5D%5C%5C%0A(3)%5Cforall%20x%5Cin%20f(M)%5B-x%5Cin%20f(M)%5C%5C%0A%5Cforall%20a%5Cin%20R%5Cforall%20x%5Cin%20f(M)%5Bax%5Cin%20f(M)%5D)
![(1)0\in f^{-1}(0)\\
(2)\forall x,y\in f^{-1}(0)[x+y\in f^{-1}(0)]\\
(3)\forall x\in f^{-1}(0)[-x\in f^{-1}(0)]\\
(4)\forall a\in R\forall x\in f^{-1}(0)[ax\in f^{-1}(0)]](https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=(1)0%5Cin%20f%5E%7B-1%7D(0)%5C%5C%0A(2)%5Cforall%20x%2Cy%5Cin%20f%5E%7B-1%7D(0)%5Bx%2By%5Cin%20f%5E%7B-1%7D(0)%5D%5C%5C%0A(3)%5Cforall%20x%5Cin%20f%5E%7B-1%7D(0)%5B-x%5Cin%20f%5E%7B-1%7D(0)%5D%5C%5C%0A(4)%5Cforall%20a%5Cin%20R%5Cforall%20x%5Cin%20f%5E%7B-1%7D(0)%5Bax%5Cin%20f%5E%7B-1%7D(0)%5D)
%5Csimeq%20M%2FKer%20f)
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