Base (module)

The concept of the basis of a module is a generalization of the concept of the basis of a vector space in the mathematical branch of algebra . As with these, a base of a module is defined as a linearly independent generating system; In contrast to vector spaces, however, not every module has a basis.

definition

A system of elements of a module over a ring with one element defines a map ${\displaystyle \{x_{i}\mid i\in I\}}$ ${\displaystyle M}$ ${\displaystyle R}$

${\displaystyle \xi \colon R^{(I)}\longrightarrow M}$

from the direct sum of copies from after that from the illustrations ${\displaystyle R}$ ${\displaystyle M}$

${\displaystyle R\to M,\quad 1\mapsto x_{i}}$

is induced.

• Is injective , it means linearly independent . ${\displaystyle \xi }$ ${\displaystyle \{x_{i}\mid i\in I\}}$
• If surjective , then a generating system is called . ${\displaystyle \xi }$ ${\displaystyle \{x_{i}\mid i\in I\}}$
• If bijective , then a base is called from . ${\displaystyle \xi }$ ${\displaystyle \{x_{i}\mid i\in I\}}$ ${\displaystyle M}$

A basis is therefore a linearly independent generating system.

properties

The linear independence of is equivalent to the fact that the 0 can only be represented as the trivial linear combination: ${\displaystyle \{x_{i}\mid i\in I\}}$

${\displaystyle \sum a_{i}x_{i}=0\quad \Longrightarrow \quad a_{i}=0\ \mathrm {f{\ddot {u}}r\ alle} \ i\in I.}$

If a set is linearly dependent, it generally does not follow from this - in contrast to the case of vector spaces - that one of the elements can be represented as a linear combination of the others. This has the following consequences:

• In general, a linearly independent subset cannot be added to a basis.
• A maximally linearly independent subset is generally not a basis.
• A minimal generating system is generally not a basis.

Consider the module as an example : The system {2} is maximally linearly independent, the system {2,3} is a minimal generating system, neither of the two is a basis. ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$

A module over a ring with a single element has a base if and only if it is free . The term free module is a generalization of the basic existence on modules whose base ring does not necessarily have a single element. Each sub-module of a free module is free again via main ideal rings .

Inductive calculation of a base

If a free module is over a main ideal ring and a sub-module of , then a basis of inductive can be calculated: ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle N}$ ${\displaystyle M}$ ${\displaystyle N}$

Be a base of , consider . ${\displaystyle \{m_{1},\dotsc ,m_{n}\}}$ ${\displaystyle M}$ ${\displaystyle N_{i}=N\cap \langle m_{1},\dotsc ,m_{i}\rangle }$

The ideal is generated by the ring element and it is ${\displaystyle \{r\in R:\exists m\in N_{i+1}{\text{ mit }}m=m'+r\cdot m_{i+1}{\text{ und }}m'\in \langle m_{1},\dotsc ,m_{i}\rangle \}}$ ${\displaystyle a_{i+1}}$

${\displaystyle n_{i+1}=m'+a_{i+1}\cdot m_{i+1}\in N_{i+1}{\text{ mit }}m'\in \langle m_{1},\dotsc ,m_{i}\rangle }$,

then applies . ${\displaystyle N_{i+1}=N_{i}\oplus R\cdot n_{i+1}}$

example

Let be a module and the sub-module is defined by . ${\displaystyle M=\mathbb {Z} ^{3}=\langle (1,0,0),(0,1,0),(0,0,1)\rangle }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle N:=\{z\in \mathbb {Z} ^{3}:2z_{1}+3z_{2}+4z_{3}=0\land 5{\text{ teilt }}z_{2}\}}$

A basis of can now be calculated as follows: ${\displaystyle N}$

${\displaystyle N_{1}=N\cap \langle (1,0,0)\rangle =\{z\in \mathbb {Z} ^{3}:2z_{1}=0\}=\{(0,0,0)\}}$

${\displaystyle N_{2}=N\cap \langle (1,0,0),(0,1,0)\rangle =\{z\in \mathbb {Z} ^{3}:2z_{1}+3z_{2}=0\land 5\vert z_{2}\}}$

We are now looking for the smallest positive that satisfies the above equation. ${\displaystyle z_{2}}$

${\displaystyle \Rightarrow N_{2}=\langle (-15,10,0)\rangle }$

${\displaystyle N_{3}=N\cap \langle (1,0,0),(0,1,0),(0,0,1)\rangle =N}$

We're looking for the smallest positive that satisfies the equation. ${\displaystyle z_{3}}$

${\displaystyle \Rightarrow N_{3}=N_{2}\oplus \langle (-2,0,1)\rangle }$

We have found a base . ${\displaystyle N=\langle (-15,10,0),(-2,0,1)\rangle }$

Examples

ℤ as a ℤ module

Let it be the Abelian group of whole numbers as a module over the ring of whole numbers. Then ${\displaystyle M=\mathbb {Z} }$

• ${\displaystyle \{2\}}$ a maximal linearly independent subset, but no generating system.
• ${\displaystyle \{2,3\}}$ a minimal generating system, but not linearly independent.

The only bases of are and . ${\displaystyle M}$ ${\displaystyle \{1\}}$ ${\displaystyle \{-1\}}$

Grid in ℝ ⁿ as a ℤ module

Lattice with basis vectors and ${\displaystyle b_{1}=({\tfrac {2}{3}},{\tfrac {1}{3}})}$ ${\displaystyle b_{2}=({\tfrac {1}{3}},-{\tfrac {1}{3}})}$

Let there be linearly independent vectors of the Euclidean vector space . Then it is called the module ${\displaystyle b_{1},b_{2},\ldots ,b_{m}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {Z} }$

${\displaystyle \Gamma :=\langle b_{1},\dots ,b_{m}\rangle _{\mathbb {Z} }:=\left\{\left.\textstyle \sum \limits _{i=1}^{m}g_{i}b_{i}\ \right|\,g_{i}\in \mathbb {Z} \right\}}$

a lattice with a base of rank . ${\displaystyle \{b_{1},b_{2},\ldots ,b_{m}\}}$ ${\displaystyle m}$

Grids in play a central role in the theory of elliptic functions and elliptic curves , grids in are related to complex Tori and Abelian varieties . ${\displaystyle \mathbb {C} =\mathbb {R} ^{2}}$ ${\displaystyle \mathbb {C} ^{g}=\mathbb {R} ^{2g}}$