Answer to Question #135136 in Abstract Algebra for Ram

Question #135136
Using Fundamental theorem of homomorphism to prove that the rings R^2 and R^2/R^4 are isomorphic.
1
Expert's answer
2020-09-28T12:07:59-0400
"Solution"

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)"

.

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


"R_2\/R_4\\cong \\phi(R_2)"


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