Topics in algebra Chapter IV: Commutative rings and modules I - 1

Problem set
1. Find a formula for the inverse of a nonzero element a0 +a1i+a2j +a3k
in the division ring of real quaternions.
2. Let R be a commutative ring with identity; then the set of all units of
R form a multiplicative subgroup of R.
3. Let R be a commutative ring; then the set of all nilpotent elements of
R is an ideal of R.
4. Let R be a ring and I be a left ideal of R then the set annR(I) = {r ∈
R | ra = 0∀a ∈ I} is an ideal of R.
5. Let R be a commutative ring and I be an ideal of R. Then the set
rad(I) = √ I = {r ∈ R | r n ∈ I for some n} is an ideal of R.
6. Let R be a ring and I be an ideal of R. Then the sets I : R = {r ∈
R | ar ∈ I∀a ∈ R} and 0 : I = {r ∈ R | ar ∈ 0∀a ∈ I} are ideals of R.
7. Let S be the ring of all 2 × 2 matrices over a field F . Prove the following:
a 0
(a) The center of S consists of all matrices of the form .
0 a
(b) The center of S is not an ideal of S.
What is the center of Mn(F )?
8. A ring R such that a 2 = a for all a ∈ R is called a Boolean ring. Prove
that every Boolean ring is commutative and a + a = 0 for all a ∈ R.
9. Let G = Z ⊕ Z. Prove that End G is a noncommutative ring with
identity.
10. Let G = {1, −1, i, −i, j, −j, k, −k} be the subset of the division ring
of real quaternions. Prove that G is a group.
11 Prove that a ring R with identity is a division ring if and only if R has
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