Left and right (algebra)
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|Left multiplication to s and right multiplication to t. An abstract notation without any specific sense.|
In algebra, the terms left and right denote the order of a binary operation (usually, but not always called "multiplication") in non-commutative algebraic structures. A binary operation ∗ is usually written in the infix form:
- s ∗ t
The argument s is placed on the left side, and the argument t is on the right side. Even if the symbol of the operation is omitted, the order of s and t 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.
Binary operation as an operator
- Rt(s) = s ∗ t,
depending on t as a parameter. It is the family of right operations. Similarly,
- Ls(t) = s ∗ t
defines the family of left operations parametrized with s.
Left and right modules
|Left module||Right module|
|s(x + y) = sx + sy
(s1 + s2)x = s1x + s2x
s(tx) = (s t)x
|(x + y)t = xt + yt
x(t1 + t2) = xt1 + xt2
(xs)t = x(s t)
The distinction is not purely syntactical because implies two different associativity rules (the lowest row in the table) 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 obvious associativity condition on them.