In algebra (a branch of mathematics), many algebraic structures have trivial , that is, simplest objects . Like sets, they consist of one element , denoted by the symbol “ 0 ”, and the object itself as “ {0} ”, or simply “0” depending on the context (for example, in exact sequences ). Objects corresponding to trivial cases are important for unification of reasoning: for example, it is more convenient to say that “solutions of the equation T x = 0 always make up a linear space” rather than making a reservation “... or the set { 0 }”.
The most important of these facilities are:
- A trivial group , the simplest of groups .
- It is also the simplest of Abelian groups , and all of the following objects inherit its structure, understood as addition .
- A trivial ring , the simplest of rings .
- Zero (trivial, or empty generated ) module , the simplest of modules over a given ring R ).
- Zero (or zero-dimensional ) linear space over the field R , the simplest of linear spaces.
- Zero algebra , the simplest of algebras over a ring or over a field R.
In the last three cases, scalar multiplication is defined as κ0 = 0 , where κ ∈ R.
Every zero algebra is also trivial like a ring. Zero algebra over a field is a zero linear space, and over a ring it is a zero module.
Content
Interpretation Using Category Theory
From the point of view of category theory , a trivial object is a terminal , and sometimes (depending on the definition of morphism ) zero (that is, both terminal and initial ) object.
The trivial object is unique up to isomorphism .
The terminal nature of the trivial object means that the morphism A → {0} exists and is unique for any object A in the category. This morphism maps every element of the object A to 0 .
| 2 ↕ | = | [ ] | ‹0 | ||
| ↔ one | ^ 0 | ↔ one | |||
| An element of zero space written as an empty column vector (on the right) is multiplied by an empty 2 × 0 matrix to obtain a 2-dimensional zero vector (on the left). Matrix multiplication rules are observed. | |||||
In the categories Rng (rings without a mandatory unit), R are Mod and Vect R , the trivial ring, zero module and space are respectively zero objects. The zero object is by definition started, that is, the morphism {0} → A exists and is unique for any object A in the category. This morphism maps 0 , the only element of the object {0} , to zero 0 ∈ A. This is a monomorphism , and its image (a submodule / subspace in A generated by zero elements ) is isomorphic {0}.
Structures with Unit
In structures with a unit (a neutral element of multiplication), things are not so simple. When the definition of morphism in a category requires their preservation, the trivial object is either only terminal (but not initial) or does not exist at all (for example, when the definition of structure requires the inequality 1 ≠ 0 ).
In the Ring category of unit rings, the ring of integers Z is the starting object, not {0}.
See also
- Empty semigroup
- Trivial algebra
Links
- David Sharpe. Rings and factorization. - Cambridge University Press , 1987. - P. 10 : trivial ring . - ISBN 0-521-33718-6 .
- Barile, Margherita. Trivial Module on Wolfram MathWorld .
- Barile, Margherita. Zero Module on Wolfram MathWorld .