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# We fix a field F. Let R be a ring containing F.

We fix a field F. Let R be a ring containing F. We say that R is an algebra over F if addition and multiplication in R restrict to addition and multiplication in F and the element 1 is in F if the element 1 is in R. Let R be and algebra over F.1. Suppose that φ1 and φ2 are distinct homomorphisms F[x] --> R such that φ1(c) = φ2(c) = c for every c in the field. Show that φ1(x) does not equal φ2(x).2. If r is any element of R, show that there is a homomorphism φ: F[x] --> R such that φ(x) = r and φ1(c) = φ2(c) = φ1 for every c in F.3. Show that there is a bijection between the set R and the set of homomophisms φ: F[X] --> R such that φ1(c) = φ2(c) = c for every c in F