If the operation denoted is not commutative, there is a distinction between left-distributivity and right-distributivity:
In either case, the distributive property can be described in words as:
To multiply a sum (or difference) by a factor, each summand (or minuend and subtrahend) is multiplied by this factor and the resulting products are added (or subtracted).
If the operation outside the parentheses (in this case, the multiplication) is commutative, then left-distributivity implies right-distributivity and vice versa, and one talks simply of distributivity.
One example of an operation that is "only" right-distributive is division, which is not commutative:
Multiplying sums can be put into words as follows: When a sum is multiplied by a sum, multiply each summand of a sum with each summand of the other sum (keeping track of signs) then add up all of the resulting products.
In the following examples, the use of the distributive law on the set of real numbers is illustrated. When multiplication is mentioned in elementary mathematics, it usually refers to this kind of multiplication. From the point of view of algebra, the real numbers form a field, which ensures the validity of the distributive law.
First example (mental and written multiplication)
During mental arithmetic, distributivity is often used unconsciously:
Thus, to calculate in one's head, one first multiplies and and add the intermediate results. Written multiplication is also based on the distributive law.
Second example (with variables)
Third example (with two sums)
Here the distributive law was applied twice, and it does not matter which bracket is first multiplied out.
Here the distributive law is applied the other way around compared to the previous examples. Consider
Since the factor occurs in all summands, it can be factored out. That is, due to the distributive law one obtains
In standard truth-functional propositional logic, distribution in logical proofs uses two valid rules of replacement to expand individual occurrences of certain logical connectives, within some formula, into separate applications of those connectives across subformulas of the given formula. The rules are
Distributivity is a property of some logical connectives of truth-functional propositional logic. The following logical equivalences demonstrate that distributivity is a property of particular connectives. The following are truth-functional tautologies.
A ring has two binary operations, commonly denoted and and one of the requirements of a ring is that must distribute over
A lattice is another kind of algebraic structure with two binary operations,
If either of these operations distributes over the other (say distributes over ), then the reverse also holds ( distributes over ), and the lattice is called distributive. See also Distributivity (order theory).
A 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.
Failure of one of the two distributive laws brings about near-rings and near-fields instead of rings and division rings respectively. The operations are usually configured to have the near-ring or near-field distributive on the right but not on the left.
Rings and distributive lattices are both special kinds of rigs, which are generalizations of rings that have the distributive property. For example, natural numbers form a rig.
In several mathematical areas, generalized distributivity laws are considered. This may involve the weakening of the above conditions or the extension to infinitary operations. Especially in order theory one finds numerous important variants of distributivity, some of which include infinitary operations, such as the infinite distributive law; others being defined in the presence of only one binary operation, such as the according definitions and their relations are given in the article distributivity (order theory). This also includes the notion of a completely distributive lattice.
In the presence of an ordering relation, one can also weaken the above equalities by replacing by either or Naturally, this will lead to meaningful concepts only in some situations. An application of this principle is the notion of sub-distributivity as explained in the article on interval arithmetic.
In the context of a near-ring, which removes the commutativity of the additively written group and assumes only one-sided distributivity, one can speak of (two-sided) distributive elements but also of antidistributive elements. The latter reverse the order of (the non-commutative) addition; assuming a left-nearring (i.e. one which all elements distribute when multiplied on the left), then an antidistributive element reverses the order of addition when multiplied to the right: 
In the study of propositional logic and Boolean algebra, the term antidistributive law is sometimes used to denote the interchange between conjunction and disjunction when implication factors over them: