19.2.6. Proof of Invertible Elements Forming a Subgroup

Try these first

1. 1.

   Define what it means for an element to be invertible in a ring.

   Hint

   Think about the identity element in multiplication.

2. 2.

   Give an example of an invertible element in ℤ₃.

   Hint

   Check what happens when you multiply them with each other.

3. 3.

   What is an invertible element?

   • An element with no inverse
   • An element that has a multiplicative inverse
   • An element that is zero

   Hint

   Think back to the definition we just discussed.

4. 4.

   True or False: The closure property states that multiplying elements in a set retains the result within the same set.

   • True
   • False

   Hint

   Recall the definition of closure we talked about.

5. 5.

   Prove that the product of any two invertible elements in ℤₙ results in an invertible element in the same ring.

   Hint

   Focus on properties of inverses and closure.

6. 6.

   Find all invertible elements in the ring ℤ₁₀ and justify your reasoning.

   Hint

   Use the definition of co-primality to help identify these numbers.