+−⟲ Monoid Semigroup Group Field + identity element + inverses Abelian Group + commutativity + multiplicative identity + commutative multiplication + commutative multiplication + invertibility (non-zero) Division Ring (Skew Field) Commutative Ring Ring Rng (without Unity) Magma + associativity Set + binary operation Semiring + additive inverses + second operation (multiplication) + associative& distributive + invertibility (non-zero) Integral Domain + no zero-divisors + invertibility (non-zero) Commutative Monoid + commutativity + second operation (multiplication) + associative& distributive + the absorption property of zero