100(ii) Symmetric: Ifϕ:A→Ris an isomorphism, then part (i)shows that its inverse is an isomorphismR→A.(iii) Transitive: IfAϕ→Bθ→Cbe ring homomorphisms, then part (ii) shows that their com-posite is an isomorphismA→C.3.45LetRbe a commutative ring and letF(R)be the commutative ring of allfunctionsf:R→R.(i)Show thatRis isomorphic to the subring ofF(R)consisting ofall the constant functions.Solution.Ifr∈R, letεrdenote the constant functionR→Rsendinga→rfor alla∈R. It is routine to check thatϕ:R→F(R), given byr→εr, is an injective homomorphism with thedesired image.(ii)Iff(x)=a0+a1x+· · ·+anxn∈R[x], letf:R→Rbe definedbyf(r)=a0+a1r+ · · · +anrn[thus,fis the polynomialfunction associated tof(x)]. Show that the functionϕ:R[x] →F(R), defined byϕ:f(x)→f, is a ring homomorphism.Solution.Absent.(iii)Show that ifR=Fp, wherepis a prime, thenxp−x∈kerϕ.Solution.
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