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Mathematics 366 29) Homework (due Oct. 7) A. Hulpke Let R = Z and I =< 15, 24 >R ⊲ R. Show that I is a principal ideal and describe its members. 30) Let R be the ring of all functions defined on the set of real numbers with pointwise addition/multiplication. (e.g. ( f + g)(x) = f (x) + g(x), ( f ⋅ g)(x) = f (x) ⋅ g(x)). a) Let S be the set of all differentiable functions. Show that S ≤ R but S ⊲/ R. b) Let T = { f ∈ R ∣ f (1) = 0}. Then T ⊲ R (you do not need to show this again). Is T a principal ideal? c) Determine a ring homomorphism φ defined on R such that ker φ = T. a b a b ) ∣ a, b, c ∈ Q} and φ∶ R → Q, ( ) ↦ a. Prove that φ is a ring homo0 c 0 c morphism. What is its kernel? 31) Let R = {( 32) Let R = {( φ∶ R → C, ( a b ) ∣ a, b, c ∈ R}. Then (you don’t need to show this) R ≤ M2×2 (R). Let −b a a b ) ↦ a + bi. Show that φ is a ring isomorphism. −b a 33) a) Let φ∶ R → S be a ring homomorphism. Show that φ is one-to-one if and only if ker φ = {0}. b) Now assume that R is a field and φ∶ R → R a ring homomorphism such that φ(1) =/ 0. Show that φ is a ring isomorphism. 34∗ ) Let R, S be rings with one and φ∶ R → S a ring homomorphism that is onto. a) Show that φ(1R ) = 1S , i.e. the image of the one in R is the one in S. b) Show that if r ∈ R is a unit, that the image φ(r) must be a unit in S. c∗ ) Give an example that the statements under a),b) are wrong, if φ is not onto. Problems marked with a ∗ are bonus problems for extra credit. For the midterms/final you may bring a single page of unrestricted notes. It must be handwritten by yourself and carry your name. You may only consult your own notes during the exam.