Question #138067
Use fundamental theorem of homomorphism to prove that the ring R^2 and R^4/R^2 are isomorphic.
1
Expert's answer
2020-10-13T18:30:55-0400

ϕ:R4R2\phi: R^4\longrightarrow R^2 is given by (a,b,c,d)(a,b)(a,b,c,d)\mapsto (a,b) This map is trivially homomorphism. This map is clearly surjective and kernel is given by a=b=0.{a=b=0}. So by homomorphism theorem R4/KerϕR2.R^4 /Ker \phi\cong R^2. Now we need to show KerϕR2.Ker \phi\cong R^2. This is given by the map (0,0,x,y)(x,y)(0,0,x,y)\mapsto (x,y) . This map is clearly surjective and its kernel is zero and hence bijective. Homomorphism is trivial.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS