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Abstract Algebra: Rings and Fields

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What is a ring in abstract algebra?

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A
A non-empty set with two binary operations, addition and multiplication, that satisfy certain properties.
💡 Explanation:

A ring is a non-empty set R equipped with two binary operations, addition (+) and multiplication (x), that satisfy the following properties:

  1. Associativity of addition: For all a, b, c in R, (a + b) + c = a + (b + c).
  2. Commutativity of addition: For all a, b in R, a + b = b + a.
  3. Existence of additive identity: There exists an element 0 in R such that for all a in R, a + 0 = a.
  4. Existence of additive inverse: For each a in R, there exists an element -a in R such that a + (-a) = 0.
  5. Associativity of multiplication: For all a, b, c in R, (a x b) x c = a x (b x c).
  6. Distributivity of multiplication over addition: For all a, b, c in R, a x (b + c) = (a x b) + (a x c).
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