Mathematics 366
Homework (due Oct. 14)
A. Hulpke
35) Let R = Q[x] and A = { f ∈ R ∣ f (1) = 0}. Show that the set of polynomials a ⋅ x (for a ∈ Q)
is a set of representatives of the cosets of A in R. (You need to show that for a =/ b we have that
A + ax =/ A + bx and that for every f ∈ R we have that there is an a such that A + f = A + ax.)
36) Let R = Z12 and I = ⟨4⟩ ⊲ R. Determine the cosets of I in R.
37) Let φ∶ R → S be a ring homomorphism and A = ker φ ⊲ R. Show that for r, s ∈ R we have that
φ(r) = φ(s) if and only if r and s are in the same coset of A, i.e. if and only if A + r = A + s.
38) Let R = Q[x] and I =< x 2 − 1 >R ⊲ R.
a) Describe the cosets of I in R.
b) Show that the quotient ring is not an integral domain.
39∗ )
Show that on the set R of real numbers the relation ∼, defined by
a ∼ b ∶⇔ ∣a − b∣ ∈ Z
is an equivalence relation.
Is it also an equivalence relation on C?
40)
Let R = Q[x] and A = ⟨x 2 + x + 1⟩ ⊲ R. Find an inverse to A + (x + 1) in R/A.
41) Let R be a ring and A ⊲ R. Show that R/A is commutative if and only if rs−sr ∈ A for all r, s ∈ R.
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.