# Distributivity

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In algebra, distributivity is a property of two binary operations which generalises the relationship between addition and multiplication in elementary algebra known as "multiplying out". For these elementary operations it is also known as the distributive law, expressed as

$a \times (b + c) = (a \times b) + (a \times c)$

Formally, let $\otimes$ and $\oplus$ be binary operations on a set X. We say that $\otimes$ left distributes over $\oplus$, or is left distributive, if

$a \otimes (b \oplus c) = (a \otimes b) \oplus (a \otimes c) \,$

and $\otimes$ right distributes over $\oplus$, or is right distributive, if

$(b \oplus c) \otimes a = (b \otimes a) \oplus (c \otimes a) . \,$

The laws are of course equivalent if the operation $\otimes$ is commutative.

## Examples

• In a ring, the multiplication distributes (both left and right) over the addition.
• In a vector space, multiplication by scalars distributes over addition of vectors. (Note however that here the two multipliers are of different type: one scalar, the other vector.)
• There are three closely connected examples where each of two operations distributes over the other: