(R, ⊕, 0, ⊖, ⊙, 1) forms a ring :⇔ (R, ⊕, 0, ⊖) forms an abelian group and (R, ⊙, 1) forms a monoid and ⊙ is distributive over ⊕ and 0 is an absorbing element for ⊙ and ∀ a, b ∈ R : a ⊙ ⊖(b) = ⊖(a) ⊙ b = ⊖(a ⊙ b) ⇔ (R, ⊕, 0, ⊖) forms an abelian group and (R, ⊙, 1) forms a monoid and ⊙ is distributive over ⊕ ⇔ (R, ⊕, 0, ⊙, 1) forms a semiring and ∀ a ∈ R : a ⊕ ⊖(a) = ⊖(a) ⊕ a = 0 Equivalence. No proof.
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