19.2.5. The Set U(ℝ)

Try these first

1. 1.

   List the invertible elements in ℤ_7.

   Hint

   Consider which numbers are coprime to 7.

2. 2.

   What does it mean for an element to be invertible in a ring?

   Hint

   Think of multiplication properties.

3. 3.

   What is U(ℝ)?

   • The set of all integers
   • The set of all invertible elements in a ring
   • The set of all real numbers

   Hint

   Recall the definition of invertible elements.

4. 4.

   True or False: Every element in a ring is invertible.

   • True
   • False

   Hint

   Think about the examples of rings provided.

5. 5.

   Determine the elements of U(ℤ_{15}) and prove they are invertible.

   Hint

   Check each number to see if it shares any factors with 15.

6. 6.

   Prove that U(ℤ_N) is a group under multiplication.

   Hint

   Utilize the definitions of groups and focus on showing properties.