Direct sum of Abelian groups

The term direct sum of Abelian groups generalizes the concept of the direct sum of vector spaces . It is of great importance for the theory of Abelian groups . If a group can be broken down into a direct sum, the structure of the group is reduced to simpler groups. New groups can be formed from the direct summands. Most structure theorems make a statement about a direct decomposition of groups.

Definitions

• The Abelian group is said to be a direct sum of two subgroups , if the following two conditions are met. ${\displaystyle A}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$

1. ${\displaystyle A_{1}+A_{2}=A}$.
2. ${\displaystyle A_{1}\cap A_{2}=0}$.

In this case it is written . Here referred to the subgroup that only the identity element contains. ${\displaystyle A=A_{1}\oplus A_{2}}$ ${\displaystyle 0}$ ${\displaystyle 0}$

• A subgroup is a direct summand if there is a subgroup are with: . In this case the complement of . ${\displaystyle A_{1}\hookrightarrow A}$ ${\displaystyle A_{2}}$ ${\displaystyle A=A_{1}\oplus A_{2}}$ ${\displaystyle A_{2}}$ ${\displaystyle A_{1}}$
• ${\displaystyle A}$means directly indecomposable if and are the only direct summands of . ${\displaystyle A}$ ${\displaystyle 0}$ ${\displaystyle A}$
• Be a family of subgroups of . The group is called the direct sum of when the following conditions are met. ${\displaystyle (A_{i}|i\in I)}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle (A_{i}|i\in I)}$

1. ${\displaystyle \sum _{i\in I}A_{i}=A}$. The family creates . ${\displaystyle (A_{i}|i\in I)}$ ${\displaystyle A}$
2. For each applies: . ${\displaystyle i\in I}$ ${\displaystyle A_{i}\cap \sum _{j\neq i}A_{j}=0}$

It is written: . Or if . ${\displaystyle A=\bigoplus \limits _{i\in I}A_{i}}$ ${\displaystyle A=A_{1}\oplus \dots \oplus A_{n}}$ ${\displaystyle I=\{1,\dots ,n\}}$

Explanations, simple sentences

• They are subgroups of the Abelian group . Then the following statements are equivalent: ${\displaystyle B,C}$ ${\displaystyle A}$
  • It is . ${\displaystyle A=B\oplus C}$
  • Each can be clearly written as with . ${\displaystyle a\in A}$ ${\displaystyle a=b+c}$ ${\displaystyle b\in B,c\in C}$
  • It is and follows from with . ${\displaystyle A=B+C}$ ${\displaystyle 0=b+c}$ ${\displaystyle b\in B,c\in C}$ ${\displaystyle b=0=c}$
• If , then the two endomorphisms and have the following property: and . It is the identity . ${\displaystyle A=B\oplus C}$ ${\displaystyle \pi _{B}\colon A\ni b+c\mapsto b\in A}$ ${\displaystyle \pi _{C}\colon a\ni b+c\mapsto c\in A}$ ${\displaystyle \pi _{B}^{2}=\pi _{B},\pi _{C}^{2}=\pi _{C},\pi _{C}\pi _{B}=\pi _{B}\pi _{C}=0}$ ${\displaystyle 1_{A}=\pi _{B}+\pi _{C}}$ ${\displaystyle 1_{A}}$ ${\displaystyle A}$

Homomorphisms provide a way of identifying and recognizing direct summands:

• Be homomorphisms. Then: ${\displaystyle A{\overset {f}{\longrightarrow }}B{\overset {g}{\longrightarrow }}C}$

1. ${\displaystyle gf}$is a monomorphism and is a monomorphism. ${\displaystyle \iff \operatorname {Bild} (f)\cap \operatorname {Kern} (g)=0}$ ${\displaystyle f}$
2. If there is an epimorphism , then is . ${\displaystyle gf}$ ${\displaystyle \operatorname {Bild} (f)+\operatorname {Kern} (g)=B}$
3. If there is an isomorphism , then is . ${\displaystyle gf}$ ${\displaystyle \operatorname {Bild} (f)\oplus \operatorname {Kern} (g)=B}$

• The following statements are equivalent for a subgroup : ${\displaystyle B\hookrightarrow A}$

1. ${\displaystyle B}$is a direct summand in . ${\displaystyle A}$
2. There is an endomorphism with: and . ${\displaystyle \pi \colon A\to A}$ ${\displaystyle \pi ^{2}=\pi }$ ${\displaystyle \pi (A)=B}$
3. If the inclusion map is, there is a homomorphism with . ${\displaystyle \iota \colon B\to A}$ ${\displaystyle \pi \colon A\to B}$ ${\displaystyle \pi \iota =1_{B}}$

Examples

• ${\displaystyle 0}$ is a direct summand in every group.
• Let it be the cyclic group with the associated addition. Be it . Then is . There are and subsets of . Their average is and their sum is . It is for example . ${\displaystyle A}$ ${\displaystyle A=\mathbb {Z} /6\mathbb {Z} =\{0,1,2,3,4,5\}}$ ${\displaystyle U=\{0,3\},V=\{0,2,4\}}$ ${\displaystyle A=U\oplus V}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle A}$ ${\displaystyle 0}$ ${\displaystyle A}$ ${\displaystyle 3+4=1{\bmod {6}}}$
• The groups of whole numbers and rational numbers are indecomposable. If a prime number , it is directly indecomposable. ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle p}$ ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$
• If the Abelian group has a largest subgroup , then it is directly indivisible. If a prime number, then has the largest subgroup . So it cannot be decomposed directly. ${\displaystyle \neq A}$ ${\displaystyle A}$ ${\displaystyle p}$ ${\displaystyle A=\mathbb {Z} /p^{n}\mathbb {Z} }$ ${\displaystyle p\mathbb {Z} /p^{n}\mathbb {Z} \neq A}$ ${\displaystyle A}$
• Are coprime integers, then is . ${\displaystyle a,b}$ ${\displaystyle \mathbb {Z} /(a\cdot b)\mathbb {Z} =a\mathbb {Z} /(a\cdot b)\mathbb {Z} \oplus b\mathbb {Z} /(a\cdot b)\mathbb {Z} }$
• The last example has a strong generalization: be a group and be with . In addition, be coprime with . Then is . ${\displaystyle A}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle A\cdot n=0}$ ${\displaystyle n=r\cdot s}$ ${\displaystyle r,s}$ ${\displaystyle A=A\cdot r\oplus A\cdot s}$
• Is , so is , where is. The complement of is by no means clearly determined. For example, it is also for everyone . ${\displaystyle \mathbb {Z} ^{2}=\{(z_{1},z_{2})|z_{1},z_{2}\in \mathbb {Z} \}}$ ${\displaystyle \mathbb {Z} ^{2}={\vec {e_{1}}}\mathbb {Z} \oplus {\vec {e_{2}}}\mathbb {Z} }$ ${\displaystyle {\vec {e_{1}}}=(1,0),{\vec {e_{2}}}=(0,1)}$ ${\displaystyle {\vec {e_{1}}}\mathbb {Z} }$ ${\displaystyle \mathbb {Z} ^{2}={\vec {e_{1}}}\mathbb {Z} \oplus (z,1)\mathbb {Z} }$ ${\displaystyle z\in \mathbb {Z} }$
• The last example is more general. Be a natural number. the set of - tuples with components . Next is the tuple that has one here and in other places . Then is . ${\displaystyle n\geq 1}$ ${\displaystyle \mathbb {Z} ^{n}:=\{(z_{1},\dots ,z_{n})|z_{i}\in \mathbb {Z} \}}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle {\vec {e_{i}}}}$ ${\displaystyle i}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle \mathbb {Z} ^{n}=\bigoplus \limits _{i=1}^{n}{\vec {e}}_{i}\mathbb {Z} }$
• To determine whether a subgroup is a direct summand, there is a simple criterion: ${\displaystyle \mathbb {Z} ^{2}}$

Be . Then the following statements are equivalent. ${\displaystyle {\vec {a}}=(a_{1},a_{2})\in \mathbb {Z} ^{2}}$

1. ${\displaystyle {\vec {a}}\mathbb {Z} }$is a direct summand in . ${\displaystyle \mathbb {Z} ^{2}}$
2. There is with . ${\displaystyle b_{1},b_{2}\in \mathbb {Z} }$ ${\displaystyle a_{1}\cdot b_{1}+a_{2}\cdot b_{2}=1}$

• Some grids and determinants as area
• 

  A grid with the generating vectors ${\displaystyle (1,0)\mathbb {Z} ,(0,1)\mathbb {Z} }$

• 

  The determinant as area

• 

  A grid with the generating vectors is shown ${\displaystyle (1,1)\mathbb {Z} ,(-2,-1)\mathbb {Z} }$

Property 2. of the last sentence has a geometric meaning: The subgroup is a direct summand in if and only if there is a vector such that a parallelogram of area 1 spans. ${\displaystyle {\vec {a}}\mathbb {Z} }$ ${\displaystyle \mathbb {Z} ^{2}}$ ${\displaystyle {\vec {b}}=(b_{1},b_{2})}$ ${\displaystyle {\vec {a}},{\vec {b}}}$

• The last statement can be generalized. If so the following applies: is a direct summand in if and only if the numbers have the greatest common factor . ${\displaystyle {\vec {a}}=(a_{1},\dots ,a_{n})\in \mathbb {Z} ^{n}}$ ${\displaystyle {\vec {a}}\mathbb {Z} }$ ${\displaystyle \mathbb {Z} ^{n}}$ ${\displaystyle a_{1},\dots ,a_{n}}$ ${\displaystyle 1}$

Primary groups

The following theorem makes a statement about the decomposition of torsion groups. For this purpose it is defined: Let be a prime number. The group is called -primarily if and only if everyone has a with . The sum of all -primary subgroups of a group is -primary. It is the largest primary subgroup of . It is denoted by and is called the primary component of . The following applies: ${\displaystyle p}$ ${\displaystyle A}$ ${\displaystyle p}$ ${\displaystyle a\in A}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle a\cdot p^{n}=0}$ ${\displaystyle p}$ ${\displaystyle A}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle A}$ ${\displaystyle A_{p}}$ ${\displaystyle p}$ ${\displaystyle A}$

If a torsion group is, then is . It is the direct sum of its primary components. ${\displaystyle A}$ ${\displaystyle A=\bigoplus \limits _{p{\text{ prim}}}A_{p}}$ ${\displaystyle A}$

Universal property

• Be for two subgroups and the canonical inclusions. They are equivalent: ${\displaystyle A=A_{1}+A_{2}}$ ${\displaystyle A_{1},A_{2}}$ ${\displaystyle q_{i}\colon A_{i}\hookrightarrow A}$

1. ${\displaystyle A=A_{1}\oplus A_{2}}$.
2. For every two homomorphisms there is exactly one homomorphism with for . ${\displaystyle f_{i}\colon A_{i}\to B,i\in \{1,2\}}$ ${\displaystyle f\colon A\to B}$ ${\displaystyle f\circ q_{i}=f_{i}}$ ${\displaystyle i\in \{1,2\}}$

The second statement of the theorem is the so-called universal property of the direct sum. It applies to any index set.

• Be a family of subgroups with . And be the inclusions. Then are equivalent: ${\displaystyle (A_{i}|i\in I)}$ ${\displaystyle \sum \limits _{i\in I}A_{i}=A}$ ${\displaystyle q_{i}\colon A_{i}\hookrightarrow A}$

1. It is . ${\displaystyle \bigoplus \limits _{i\in I}A_{i}=A}$
2. For each family of homomorphisms there is exactly one with . That means, the following diagram is commutative for all . ${\displaystyle f_{i}\colon A_{i}\to B}$ ${\displaystyle f\colon A\to B}$ ${\displaystyle f\circ q_{i}=f_{i}}$ ${\displaystyle i\in I}$

• Let and be two abelian groups with and . If there is exactly one with and exactly one with for every family , then and are isomorphic. ${\displaystyle (A,q_{i})}$ ${\displaystyle (S,s_{i})}$ ${\displaystyle q_{i}\colon A_{i}\to A}$ ${\displaystyle s_{i}\colon A_{i}\to S}$ ${\displaystyle f_{i}\colon A_{i}\to B}$ ${\displaystyle f\colon A\to B}$ ${\displaystyle f_{i}=f\circ q_{i}}$ ${\displaystyle g\colon S\to B}$ ${\displaystyle f_{i}=g\circ s_{i}}$ ${\displaystyle A}$ ${\displaystyle S}$

Some structure sentences

1. Theorem: If there is a homomorphism, then with and . ${\displaystyle f\colon A\to \mathbb {Z} }$ ${\displaystyle A=\operatorname {Kern} (f)\oplus a\mathbb {Z} }$ ${\displaystyle a\in A}$ ${\displaystyle a\cdot \mathbb {Z} \cong \mathbb {Z} }$
2. Theorem: Each subgroup of is the direct sum of at most cyclic subgroups. ${\displaystyle \mathbb {Z} ^{n}}$ ${\displaystyle n}$
3. Theorem: If it is torsion-free and generated by elements, then there is a monomorphism . ${\displaystyle F}$ ${\displaystyle n}$ ${\displaystyle F\to \mathbb {Z} ^{n}}$
4. Conclusion: If there is a torsion-free group generated by elements, then there is a such that is isomorphic to . ${\displaystyle F}$ ${\displaystyle n}$ ${\displaystyle k\leq n}$ ${\displaystyle F}$ ${\displaystyle \mathbb {Z} ^{k}}$
5. If finitely generated, the torsion subgroup is a direct summand of . ${\displaystyle A}$ ${\displaystyle A}$

Individual evidence

1. ^ László Fuchs: Abelian Groups. Springer, 2015, ISBN 978-3-319-19421-9 , p. 43.
2. ^ Frank W. Anderson, Kent R. Fuller: Rings and Categories of Modules. Springer, 1992, ISBN 0-387-97845-3 , p. 66.

literature

• Frank W. Anderson, Kent R. Fuller: Rings an Categories of Modules. Springer, 1992, ISBN 0-387-97845-3 .
• László Fuchs : Abelian Groups. (= Springer Monographs in Mathematics ). Springer International, 2015, ISBN 978-3-319-19421-9 .
• Friedrich Kasch : modules and rings. Teubner, Stuttgart 1977, ISBN 3-519-02211-7 .

Web links

• Since it is quite tedious to look for the evidence to the facts in the given literature together, evidence is compiled here .