## Question

###### PROBLEM 1Let R be a ring and let Hom(R,R) denote the set ofall additive maps from R to R Hom(R,R) has two operations defined on it:That is: f â‚¬ Hom(R,R) if f(a + b) = f(a) + f(b). So to check a map is in Hom(R,R), you only have to check it preserves addition:For f,g â‚¬ Hom(R, R) defineAddition by (f + g)(r) = f(r) + g(r) forallr â‚¬ R andMultiplication is composition, that is (f 0 g)(r) = f(g(r)) for allr â‚¬ RIt turns out that Hom(R,R) is a ring " under these two operations The point of

PROBLEM 1 Let R be a ring and let Hom(R,R) denote the set ofall additive maps from R to R Hom(R,R) has two operations defined on it: That is: f â‚¬ Hom(R,R) if f(a + b) = f(a) + f(b). So to check a map is in Hom(R,R), you only have to check it preserves addition: For f,g â‚¬ Hom(R, R) define Addition by (f + g)(r) = f(r) + g(r) forallr â‚¬ R and Multiplication is composition, that is (f 0 g)(r) = f(g(r)) for allr â‚¬ R It turns out that Hom(R,R) is a ring " under these two operations The point of this problem is to check a few of the ring properties: Show Hom(R,R) is closed under addition (as defined above): that is, for f,9 â‚¬ Hom(R,R), prove f + g â‚¬ Hom(R,R),i.e the sum of two ring homomorphisms is ring homomorphism Show Hom(R,R) is closed under composition (as defined aboveJ: that is, for f,9 â‚¬ Hom(R,R) prove f 9 â‚¬ Hom(R,R),ie_ the composite of two ring homomorphisms is a ring homomorphism_ Define id: R v Rby id(r) for allr â‚¬ R. Prove that id is the multiplicative identity element; that is f id = id f =f forall f â‚¬ Hom(R,R) Prove the following distributive law: f (g + h) = f 0 g + f h, for all f,g,h â‚¬ Hom(R,R) (apply both sides tor â‚¬ R and check you get the same answer):