Answer to Question #137334 in Abstract Algebra for Sohan kumar

Question #137334
Use the fundamental theorem of homomorphism to prove that rings R² and R⁴/R² are isomorphic.
1
Expert's answer
2020-10-08T16:32:52-0400

"\\phi: R^4\\longrightarrow R^2" is given by "(a,b,c,d)\\mapsto (a,b)" This map is trivially homomorphismThis map is clearly surjective and kernel is given by "{a=b=0}." So by homorphism theorem "R^4 \/Ker \\phi\\cong R^2." Now we need to show "Ker \\phi\\cong R^2." This is given by the map "(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!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS