19.2.3. Examples of Rings

Try these first

1. 1.

   What is a ring?

   Hint

   Think of a set of numbers and how operations can be performed on them.

2. 2.

   Define an abelian group.

   Hint

   Consider how the order of operations affects the result.

3. 3.

   What are the three main axioms defining a ring?

   • Closure
   • Associativity
   • Inverses
   • Closure
   • Identity
   • Distributivity
   • Closure
   • Associativity
   • Identity
   • Inverses
   • Distributivity

   Hint

   Recall the definition discussed in class.

4. 4.

   True or False: Every element in a ring has a multiplicative inverse.

   • True
   • False

   Hint

   Think about the examples presented regarding invertible elements.

5. 5.

   Prove that the integers modulo 6, Z₆, with defined addition and multiplication modulo 6 form a ring. Provide a thorough explanation for each axiom.

   Hint

   Use specific examples within the set for your proof.

6. 6.

   Using the concept of co-primality, discuss how the integers modulo 15 demonstrate which elements are invertible.

   Hint

   Find the gcd with 15 for various elements.