Theorem 1 (The Fundamental Theorem of Ring Homomorphisms): Let "(R, +_1, *_1)" and "(S, +_2, *_2)" be homomorphic rings with ring homomorphism "\\phi: R \\to S." Then "R\/ker(\\phi) \\cong \\phi (R)"
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Let "R_4 = ker(\\phi)" and let "\\Phi :R_2\/R_4 \\to \\phi(R_2)" be defined for all "(a + R_4) \\in R_2" by "\\phi (a+R_4)=\\phi(a)"
Then "\\psi" is well defined, for if "a+R_4=b+R_4 \\implies a=b+ r" for some "r \\in R_4" and so:
"\\psi (a+R_4)=\\psi ((b+r)+R_4)=\\psi((b+R_4)+R_4)=\\psi(b+R_4)"
To show that "R_2\/R_4" is an isomorphism to "\\phi (R_2)", we need to show that "\\psi" is bijective.
Let "(a+R_4), (b+R_4) \\in R_2\/R_4" and suppose that
"\\psi(a+R_4)=\\psi(b+R_4)."
Then "\\phi(a)=\\phi(b)"
So,
"\\phi(a-b)=0"
So,
"a-b \\in K,"
So, "a+R_4=b+R_4"
Hence "\\psi" is injective
Furthermore, "\\forall a \\in \\phi(R_2)" we have that "(a +I) \\in R_2\/R_4" is such that "\\psi (a + R_4)=a".
So "\\psi" is surjective.
Hence "\\psi" is bijective and so "R_2\/R_4" is proven to be an isomorphism, that is
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