This article needs additional citations for verification. (November 2012) |
| s a s b s c s d s e s f s g … | a t b t c t d t e t f t g t … |
| Left multiplication by s and right multiplication by 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.
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.
Binary operation as an operator
A binary operation ∗ may be considered as a family of unary operators through currying:
- Rt(s) = s ∗ t,
depending on t as a parameter – this is the family of right operations. Similarly,
- Ls(t) = s ∗ t
defines the family of left operations parametrized with s.
If for some e, the left operation Le is the identity operation, then e is called a left identity. Similarly, if Re = id, then e 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.
Left and right modules
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 |
|---|---|
| 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 one gets 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 associativity condition on them.[vague]
Other examples
- Left eigenvectors
- Left and right group actions
In category theory
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.
See also
- Operator associativity
External links
- . "right ideal". MathWorld.
- . "left ideal". MathWorld.
- Weisstein, Eric W. "left eigenvector". MathWorld.
This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Left and right algebra news newspapers books scholar JSTOR November 2012 Learn how and when to remove this message s a s b s c s d s e s f s g a t b t c t d t e t f t g t Left multiplication by s and right multiplication by 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 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 Binary operation as an operatorA binary operation may be considered as a family of unary operators through currying Rt s s t depending on t as a parameter this is the family of right operations Similarly Ls t s t defines the family of left operations parametrized with s If for some e the left operation Le is the identity operation then e is called a left identity Similarly if Re id then e 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 Left and right modulesOver 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 modules 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 one gets 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 associativity condition on them vague Other examplesLeft eigenvectors Left and right group actionsIn category theoryIn category theory the usage of left and right has some algebraic resemblance but refers to left and right sides of morphisms See adjoint functors See alsoOperator associativityExternal links right ideal MathWorld left ideal MathWorld Weisstein Eric W left eigenvector MathWorld
