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 over the addition.
  • In a vector space, multiplication by scalars distributes over addition of vectors.
  • There are three closely connected examples where each of two operations distributes over the other:
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