Question

Use fundamental theorem of homomorphism to prove that the ring R^2 and R^4/R^2 are isomorphic.

Answer 1

$\phi: R^4\longrightarrow R^2$ is given by $(a,b,c,d)\mapsto (a,b)$ This map is trivially homomorphism. This map is clearly surjective and kernel is given by ${a=b=0}.$ So by homomorphism 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.