… | align=right | … | - | Left multiplication to and right multiplication to . An abstract notation without any specific sense. |
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 (unspecified) of two sides.
Although the 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.
A binary operation may be considered as a family of unary operators through currying:
,depending on as a parameter – this is the family of right operations. Similarly,
defines the family of left operations parametrized with .
If for some , the left operation is the identity operation, 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 multiplication-invariant subring is a right ideal.
Over non-commutative rings, the left–right distinction is applied to modules, namely to specify the side where a scalar (module element) appears in the scalar multiplication.
| Left module !Right module | - align=center |
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A bimodule is simultaneously a left and right module, with two different scalar multiplication operations, obeying an associativity condition on them.
In category theory the usage of "left" and "right" has some algebraic resemblance, but refers to left and right sides of morphisms. See adjoint functors.