Rings (Algebraic Structure)
- is an algebraic structure that generalizes fields: multiplication need not be commutative and multiplicative inverses need not exist
Rings - Definition & Field Axioms
A ring is a set 𝐹 with two binary operations on 𝐹 called:
- addition
- multiplication
A binary operation on 𝐹 is a mapping 𝐹×𝐹 → 𝐹, that is, a correspondence that associates with each ordered pair of elements of 𝐹 a uniquely determined element of 𝐹.
Both of these operations must satisfy the ring axioms (𝑎, 𝑏, 𝑐 are arbitrary elements of the field 𝐹):
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Binary Operation 1 |
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Binary Operation 2 |
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Rings - Examples
- The set of all integers (ℤ)
- All fields are rings