Editing Distributive property (section)

==In rings and other structures==

Distributivity is most commonly found in [[semiring]]s, notably the particular cases of [[Ring (algebra)|ring]]s and [[distributive lattice]]s.

A semiring has two binary operations, commonly denoted <math>\,+\,</math> and <math>\,*,</math> and requires that <math>\,*\,</math> must distribute over <math>\,+.</math>

A ring is a semiring with additive inverses.

A [[Lattice (order)|lattice]] is another kind of [[algebraic structure]] with two binary operations, <math>\,\land \text{ and } \lor.</math> 
If either of these operations distributes over the other (say <math>\,\land\,</math> distributes over <math>\,\lor</math>), then the reverse also holds (<math>\,\lor\,</math> distributes over <math>\,\land\,</math>), and the lattice is called distributive. See also {{em|[[Distributivity (order theory)]]}}.

A [[Boolean algebra (structure)|Boolean algebra]] can be interpreted either as a special kind of ring (a [[Boolean ring]]) or a special kind of distributive lattice (a [[Boolean lattice]]). Each interpretation is responsible for different distributive laws in the Boolean algebra.

Similar structures without distributive laws are [[near-ring]]s and [[Near-field (mathematics)|near-field]]s instead of rings and [[division ring]]s. The operations are usually defined to be distributive on the right but not on the left.