notifications

Ring Theory

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semiring

A+C+0A·D

Ring

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A+C+0N+A·D

+ : Ring → Ring, · : Ring → Ring, a, b, c, 0 : Ring ⊢ a + (b + c) = (a + b) + c (associative), a + b = b + a (commutative), a + 0 = 0 + a = a (additive identity), a + (-a) = 0 (additive inverse), a (b c) = (a b) c (associative), a (b + c) = a b + a c, (a + b) c = a c + b c (distributive). Z, Q, C ≤ Ring.

Example

a, b : Ring ⊢ a + (-a) = 0 ⇒ -(-a) = a.

Commutative Ring

Wikipedia

A+C+0N+A·C·D

CommutativeRing ≤ Ring, a, b : CommutativeRing ⊢ a b = b a (commutative).

Ring with Identity

A+C+0N+A·1D

RingWithIdentity ≤ Ring, a, 1 : RingWithIdentity ⊢ a 1 = 1 a = a.

Integral domain

IntegralDomain ≤ CommutativeRing, a, b, 0 : IntegralDomain ⊢ 1 ≠ 0, a b = 0 ⇒ a = 0 or b = 0. Z, Q ≤ IntegralDomain.

Field

Wikipedia nLab

A+C+0N+A·C·1N·D

Field ≤ CommutativeRing × RingWithIdentity, a : Field ⊢ a ≠ 0 ⇒ a-1 : Field, a a-1 = 0 (multiplicative inverse). Q, C ≤ Field.

Prime Ideal

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Topological Ring

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References