Homework One
Problem I. Let A be a subset of a ring R (with 1 6= 0) and let (A)
denote the smallest ideal of R containing A, called the ideal generated
by A.
(1) Show that
\
(A) =
I.
A⊆I/R
(2) Show that
(A) = RAR := {r1 a1 r10 +r2 a2 r20 +. . .+rn an rn0 : ri , ri0 ∈ R, ai ∈ A, n ∈ N}.
(3) If R is commutative, show that
(A) = RA := {r1 a1 + r2 a2 + . . . + rn an : ri ∈ R, ai ∈ A, n ∈ N}.
Problem II.
An ideal is called principal if it is generated by a single element.
(1) Using division with remainder, show that every ideal in Z is
principal.
Hint: Let I be an ideal. Let d be the smallest element in I∩N.
Show that every element of I is divisible by d and conclude that
I = Zd.
(2) Let K be a field. Using division with remainder, show that
every ideal in the ring K[x] is principal.
Hint: Let I be an ideal. Let f be an element in I of least
degree. Show that every element of I is divisible by f , and
conclude that I = K[x]f .
Problem III.
Let F2 := Z/2Z denote a field with two elements and consider the
quotient ring R := F2 [x]/(x2 + x + 1).
(1) Prove that R has four elements: 0, 1, x and 1 + x.
(2) Write out the 4 × 4 addition table for R and prove that (R, +)
is isomorphic to the Klein 4-group Z/2Z × Z/2Z.
(3) Write out the 4 × 4 multiplication table for R and prove that
(R× , ×) is isomorphic to the cyclic group of order 3. Deduce
that R is a field.
Problem IV.
Let x4 − 16 be an element of the polynomial ring Z[x] and consider
the quotient Z[x]/(x4 − 16).
(1) Find a polynomial of degree ≤ 3 that is congruent to 7x13 −
11x9 + 5x5 − 2x3 + 3 modulo (x4 − 16).
(2) Prove that x − 2 and x + 2 are zero-divisors in Z[x]/(x4 − 16).
1
2
Problem V.
Prove that the ring M2 (R) of 2×2 matrices with real entries contains
a subring isomorphic to C.
Problem VI.
Let R be a commutative ring (with 1 6= 0). Prove that if a is nilpotent
(i.e., there is some n ∈ N such that an = 0) then 1 − ab is a unit for all
b ∈ R.
Problem VII. Let ϕ : R → S be a ring homomorphism.
(1) Prove that if J is an ideal of S, then ϕ−1 (J) is an ideal of R.
Apply this to the special case where R is a subring of S and ϕ
is the inclusion homomorphism to deduce that if J is an ideal
of S then J ∩ R is an ideal of R.
(2) Prove that if ϕ is surjective and I is an ideal of R, then ϕ(I)
is an ideal of S. Give an example to show that this fails if ϕ is
not surjective.
Problem VIII.
Prove the Lattice Isomorphism Theorem for Rings. That is, if I is
an ideal of a ring R, show that the correspondence A ↔ A/I is an
inclusion preserving bijections between the set of subrings A ⊂ R that
contain I and the set of subrings of R/I. Furthermore, show that A (a
subring containing I) is an ideal of R if and only if A/I is an ideal of
R/I.