To prove that a set is a ring, you need to verify that it satisfies certain properties that define a ring in the context of abstract algebra. A ring is a set equipped with two binary operations, typically called addition and multiplication, that satisfy specific axioms.
Here’s how to go about proving a set is a ring:
- Show that the set is non-empty: You must demonstrate that there is at least one element in the set.
- Define the operations: Clearly state the operations for addition and multiplication that you will use.
- Check the ring axioms: Verify that the set and operations satisfy the following properties:
- Closure under addition: For any two elements a and b in the set, their sum (a + b) must also be in the set.
- Associativity of addition: For any elements a, b, and c in the set, (a + b) + c must equal a + (b + c).
- Existence of additive identity: There must be an element 0 in the set such that for any element a, a + 0 = a.
- Existence of additive inverses: For every a in the set, there must exist an element -a such that a + (-a) = 0.
- Closure under multiplication: For any two elements a and b in the set, their product (a * b) must also be in the set.
- Associativity of multiplication: For any elements a, b, and c in the set, (a * b) * c must equal a * (b * c).
- Distributive properties: The multiplication operation must distribute over addition: a * (b + c) = (a * b) + (a * c).
- Check for commutativity (optional): If you want to prove the set is a commutative ring, you also need to check that for any a and b in the set, a * b = b * a.
If all these properties hold true for your set and operations, then you have successfully proven that the set forms a ring.