Left and right (algebra)


In algebra, the terms left and right denote the order of a binary operation in non-commutative algebraic structures.
A binary operation ∗ is usually written in the infix form:
The argument is placed on the left side, and the argument is on the right side. Even if the symbol of the operation is omitted, the order of and does matter unless ∗ is commutative.
A two-sided property is fulfilled on both sides. A one-sided property is related to one of two sides.
Although terms are similar, left–right distinction in algebraic parlance is not related either to left and right limits in calculus, or to left and right in geometry.

Binary operation as an operator

A binary operation may be considered as a family of unary operators through currying
depending on as a parameter. It is the family of right operations. Similarly,
defines the family of left operations parametrized with .
If for some , the left operation is identical, then is called a left identity. Similarly, if, then is a right identity.
In ring theory, a subring which is invariant under any left multiplication in a ring, is called a left ideal. Similarly, a right multiplications-invariant subring is a right ideal.

Left and right modules

Over non-commutative rings, the left–right distinction is applied to modules, namely to specify the side where a scalar appear in the scalar multiplication.
The distinction is not purely syntactical because implies two different associativity rules which link multiplication in a module with multiplication in a ring.
A bimodule is simultaneously a left and right module, with two different scalar multiplication operations, obeying an associativity condition on them.

Other examples

In category theory the usage of "left" is "right" has some algebraic resemblance, but refers to left and right sides of morphisms. See adjoint functors.