Assignment 5

Submitted on 25 September 2024.

Question 1

Have you read Sections 3.1 and 3.2? Circle one: Yes / No.
(Credit given only for an answer of “Yes”.)

Question 2

Write the units and zero divisors for each of the following rings:

  • (a) \(\mathbb{Z}\):
    • Units:
    • Zero Divisors:
  • (b) \(\mathbb{Q}\):
    • Units:
    • Zero Divisors:
  • (c) \(\mathbb{Z}_7\):
    • Units:
    • Zero Divisors:
  • (d) \(\mathbb{Z}_8\):
    • Units:
    • Zero Divisors:
  • (e) \(\mathbb{Z}_{12}\):
    • Units:
    • Zero Divisors:

Question 3

In \(\mathbb{Z}_{12}\):

  • (a) Give an example of two zero divisors whose product is zero.
  • (b) Give an example of two zero divisors whose product is nonzero.

Question 4

Define a new addition and multiplication on \( L = \mathbb{Z} \) by \( a \oplus b = a + b - 1 \) and \( a \otimes b = ab - (a + b) + 2 \). With these operations, \( L \) is a ring with identity. What is the additive identity \( 0_L \)? What is the multiplicative identity \( 1_L \)? Prove your answers are correct with two brief calculations.

\( 0_L = \)
\( 1_L = \)

Question 5

Prove that the left distributive property \( a \otimes (b \oplus c) = (a \otimes b) \oplus (a \otimes c) \) holds in the ring from the previous exercise.

Question 6

Let \( R \) be a ring with identity. Prove that the multiplicative identity is unique, as follows: Assume that \( e \) and \( f \) are both multiplicative identities; show that \( e = f \).

Question 7

Let \( R \) be a ring with identity and let \( a \) be a nonzero element of \( R \). Prove that if \( a \) has a multiplicative inverse, then it is unique, as follows: Assume that \( r \) and \( s \) are both multiplicative inverses for \( a \); show that \( r = s \).

Question 8

Suppose \( a \) and \( b \) are units in a ring \( R \) with identity. Prove that \( ab \) is a unit.
Note: You are proving that the set of units of a ring is closed under multiplication. Look up the inverse of a product of matrices from linear algebra for a hint.

Question 9

Suppose \( a \) is a unit in a commutative ring \( R \) with identity. Prove that \( a \) is not a zero divisor.

Question 10

Suppose \( R \) is a ring containing zero divisors \( a \) and \( b \). Prove that \( ab \) is either 0 or a zero divisor.