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# Let R=F[x,y] be a polynomial ring in two variables over a field F.Let M=F.Consider M as an R-module with X and Y acting as if 0.Define G:R^2 to...

Let R=F[x,y] be a polynomial ring in two variables over a field F.Let M=F.Consider M as an R-module with X and Y acting as if 0.Define G:R^2 to X*R+Y*R by G(r,s)=X*r+Y*s.Verify that N:R to R^2,T(w)=(T*w,-X*w) induces an isomorphism R=~Ker(G)