Assignment 5

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) $Z$ :
- Units:
- Zero Divisors:
(b) $Q$ :
- Units:
- Zero Divisors:
(c) $Z_{7}$ :
- Units:
- Zero Divisors:
(d) $Z_{8}$ :
- Units:
- Zero Divisors:
(e) $Z_{12}$ :
- Units:
- Zero Divisors:

Question 3

In $Z_{12}$ :

(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 $L = Z$ by $a \oplus b = a + b - 1$ and $a \otimes b = a b - (a + b) + 2$ . 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 $a b$ 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 $a b$ is either 0 or a zero divisor.