100 (ii) Symmetric: If ϕ : A → R is an isomorphism, then part (i) shows that its inverse is an isomorphism R → A . (iii) Transitive: If A ϕ → B θ → C be ring homomorphisms, then part (ii) shows that their com-posite is an isomorphism A → C . 3.45 Let R be a commutative ring and let F ( R ) be the commutative ring of all functions f : R → R . (i) Show that R is isomorphic to the subring of F ( R ) consisting of all the constant functions. Solution. If r ∈ R , let ε r denote the constant function R → R sending a 7→ r for all a ∈ R . It is routine to check that ϕ : R → F ( R ) , given by r 7→ ε r , is an injective homomorphism with the desired image. (ii) If f ( x ) = a0 + a 1 x +···+ a n x n ∈ R [ x ] , let f [ : R → R be de f ned by f [ ( r ) = a0 + a 1 r +···+ a n r n [thus, f [ is the polynomial function associated to f ( x ) ]. Show that the function ϕ : R [ x ]→ F ( R ) ,de f ned by ϕ : f ( x ) 7→ f [ , is a ring homomorphism. Solution. Absent. (iii) Show that if R = F p , where p is a prime, then x p − x ∈ ker ϕ . Solution.
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