Homework for Week 10

Part 1

  1. (Rotman 3.41i-iii; 3.29i, v-vi) True or false, with explanation.
    1. If R and S are commutative rings and f: R → S is a ring homomorphism, then f is also a homomorphism from the additive group of R to the additive group of S.
    2. If R and S are commutative rings and if f is a homomorphism from the additive group of R to the additive group of S with f(1)=1 then f is a ring homomorphism.
    3. If R and S are isomorphic commutative rings then any ring homomorphism f: R → S is an isomorphism.
    4. The sequence notation for x3-2x+5 is (5, -2, 0, 1, 0, ...).
    5. If R is a domain and f(x), g(x) are nonzero polynomials in R[x] then deg(fg) = deg(f)+deg(g).
    6. If R is a domain and f(x), g(x) are nonzero polynomials in R[x] then either f(x)+g(x) = 0 or deg (f+g) is at most the maximum of deg(f), deg(g).
  2. (Rotman 3.44i and ii)
    1. Let f: R → S be a ring isomorphism and let g: S → R be its inverse. Prove that g is an isomorphism.
    2. Show that the composition of two homomorphisms is again a homomorphism.
    3. Show that the composition of two isomorphisms is again an isomorphism.
  3. (Rotman 3.32) Let R be a domain. Prove that a polynomial f(x) is a unit in R[x] if and only if f(x) is a nonzero constant which is a unit in R. Show that (2x+1)2 = 1 in (Z/4Z)[x]. (This shows that your proof only works when R is a domain!)
  4. (Rotman 3.33) Show that if f(x) = xp - x is in (Z/pZ)[x] then its polynomial function is identically zero.
  5. (Rotman 3.38i) If f(x) = ax2p + bxp + c is in (Z/pZ)[x] prove that f'(x)=0. Can you find a condition that's necessary and sufficient for f'(x)=0 in (Z/pZ)[x]?

Part 2

  1. (Rotman 3.13) Prove that the only subring of the integers Z is Z itself.
  2. (Rotman 3.41 iv-vii) True or false, with explanations.
    1. If f is a ring homomorphism from R to a nonzero ring S, then ker f is a proper ideal in R.
    2. If I and J are ideals in a commutative ring R, then the intersection of I and J is an ideal in R, and the union of I and J is an ideal in R.
    3. If f is a ring homomorphism from R to S and if I is an ideal in R then f(I) is an ideal in S.
    4. If f is a ring homomorphism from R to S and if J is an ideal in S then the inverse image f-1(I) is an ideal in R.
  3. Prove or disprove and salvage if possible. If f(x) and g(x) are elements of (Z/nZ)[x] with f(x)g(x)=0 then either f(x)=0 or g(x)=0.
  4. List all elements of the quotient ring R/I when
    1. R= Z[x], I= 〈 x2 + 2x+1 〉
    2. R= (Z/2Z)[x], I= 〈 x4 +x2+ 1 〉
    3. R= (Z/3Z)[x], I= 〈 x3 +2x + 1 〉
    4. R= (Z/4Z)[x], I= 〈 x3 +2x + 1 〉
    For each of the previous, indicate clearly how many elements are in R/I. Any conjectures?