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A ring is a non-empty set R equipped with two binary operations, called addition (+) and multiplication (×), that satisfy the following axioms: 1. Associativity of addition: For all a, b, c ∈ R, (a + b) + c = a + (b + c). 2. Commutativity of addition: For all a, b ∈ R, a + b = b + a. 3. Existence of additive identity: There exists an element 0 ∈ R, called the additive identity, such that for all a ∈ R, a + 0 = a. 4. Existence of additive inverse: For each a ∈ R, there exists an element -a ∈ R, called the additive inverse of a, such that a + (-a) = 0. 5. Associativity of multiplication: For all a, b, c ∈ R, (a × b) × c = a × (b × c). 6. Distributivity of multiplication over addition: For all a, b, c ∈ R, a × (b + c) = (a × b) + (a × c).