Mathematics 366 Homework (due Oct. 7) A. Hulpke

Mathematics 366
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?
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.