19.2.2. Axioms of a Ring

Try these first

1. 1.

   What is the definition of a ring?

   Hint

   Think about the requirements for the set and operations.

2. 2.

   What does the closure property entail?

   Hint

   Consider what happens when you add or multiply two numbers from the set.

3. 3.

   Which of the following is NOT an axiom of a ring?

   • Associativity
   • Axiom of Choice
   • Distributive Property

   Hint

   Think about what properties specifically relate to operations in rings.

4. 4.

   True or False: All elements in a ring must have a multiplicative inverse.

   • True
   • False

   Hint

   Reflect on the definitions of rings and groups.

5. 5.

   Show that the set of integers under addition and multiplication forms a ring. Verify each axiom.

   Hint

   Use specific integer examples to verify properties.

6. 6.

   Determine if the set of even integers forms a ring under addition and multiplication. Justify your reasoning.

   Hint

   Explore the properties of even numbers in various operations.