Sub-module

The term sub-module generalizes the term sub-vector space of a vector space to a module over a ring .

definition

Let be a right module over the unitary ring . A subgroup of is called -submodule when it is completed regarding the multiplication with elements . This means that for all and all is . The term for left modules is explained accordingly. ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle U}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle U}$ ${\displaystyle R}$ ${\displaystyle u\in U}$ ${\displaystyle r\in R}$ ${\displaystyle u\cdot r\in U}$

Examples and further definitions

• Each module has the trivial sub-module and the sub-module . ${\displaystyle M}$ ${\displaystyle \{0\}}$ ${\displaystyle M}$
• If a legal module is and , it is a sub-module of . It is the cyclic sub-module created by. ${\displaystyle M}$ ${\displaystyle m\in M}$ ${\displaystyle m\cdot R:=\{m\cdot r\mid r\in R\}}$ ${\displaystyle M}$ ${\displaystyle m}$
• If the ring is a right ideal , then there is a sub- module of as a right module. ${\displaystyle I}$ ${\displaystyle R}$ ${\displaystyle I}$ ${\displaystyle R}$ ${\displaystyle R}$
• If there are sub-modules of , then there is a sub-module of . It is the smallest sub-module of that contains and . ${\displaystyle U,V}$ ${\displaystyle M}$ ${\displaystyle U+V:=\{u+v\mid u\in U,v\in V\}}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle U}$ ${\displaystyle V}$
• If there is a family of sub-modules, then is a sub-module. It is the largest sub-module contained in all of them . ${\displaystyle (U_{i}\mid i\in I)}$ ${\displaystyle \bigcap U_{i}}$ ${\displaystyle U_{i}}$
• The union of sub-modules is generally not a sub-module. So are sub-modules of but . ${\displaystyle 2\mathbb {Z} ,3\mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle 5=2+3\notin 2\mathbb {Z} \cup 3\mathbb {Z} }$

Sum of sub-modules

• If there is a right module above the ring and a family of sub- modules , then is ${\displaystyle M_{R}}$ ${\displaystyle R}$ ${\displaystyle (U_{i})_{i\in I}}$

${\displaystyle \sum _{i\in I}U_{i}=\left\{\sum _{i\in I_{e}}u_{i}\mid I_{e}\,{\text{endlich}},\ I_{e}\subset I\right\}}$

a sub-module. It is the sum of the sub-modules . ${\displaystyle (U_{i})_{i\in I}}$

• Be a subset of . Then ${\displaystyle X\subset M_{R}}$ ${\displaystyle M_{R}}$

${\displaystyle \bigcap \{V\mid V{\text{ Untermodul von }}M,X\subset V\}=\sum _{x\in X}xR}$

the smallest sub-module of which contains the set . Is ${\displaystyle M_{R}}$ ${\displaystyle X}$

${\displaystyle U=\sum _{x\in X}xR,}$

so creates the sub-module . One also says is a generating system of . ${\displaystyle X}$ ${\displaystyle U}$ ${\displaystyle X}$ ${\displaystyle U}$

• If the sub-module is generated by a finite set , it is called finitely generated. Is the crowd so is . ${\displaystyle U\subset M_{R}}$ ${\displaystyle X}$ ${\displaystyle U}$ ${\displaystyle X=\{x_{1},\dots ,x_{n}\}}$ ${\displaystyle \textstyle U=\sum _{i=1}^{n}x_{i}R}$
• A module is simply called when the only real sub-module is. A sub-module of is called maximum if the following applies to all sub-modules with : or . A module is simple precisely when every cyclic sub-module is already the same . If a real sub-module of a finitely generated module is contained in a maximum sub-module. ${\displaystyle M_{R}}$ ${\displaystyle \{0\}}$ ${\displaystyle U}$ ${\displaystyle M_{R}}$ ${\displaystyle V}$ ${\displaystyle U\subset V\subset M_{R}}$ ${\displaystyle U=V}$ ${\displaystyle V=M_{R}}$ ${\displaystyle 0\neq M_{R}}$ ${\displaystyle 0\neq U}$ ${\displaystyle M_{R}}$ ${\displaystyle U\subsetneq M_{R}}$ ${\displaystyle M_{R}}$ ${\displaystyle U}$

Inner direct sum of sub-modules

The inner direct sum of modules is defined like the inner direct sum of vector spaces. In contrast to a vector space, not every module has a basis, so that a module is usually not the inner direct sum of cyclic sub-modules.

definition

Be a family of sub-modules of the legal module and . Then the following statements are equivalent. ${\displaystyle (U_{i}\mid i\in I)}$ ${\displaystyle M}$ ${\displaystyle \textstyle V=\sum _{i\in I}U_{i}}$

• For all is . ${\displaystyle i\in I}$ ${\displaystyle \textstyle U_{i}\cap \sum _{j\neq i}U_{j}=\{0\}}$
• The following applies to all finite subsets : If , where for all , then applies to all . Each can therefore be represented in exactly one way as the sum of elements from the . ${\displaystyle I_{e}\subset I}$ ${\displaystyle \textstyle \sum _{i\in I_{e}}u_{i}=\sum _{i\in I_{e}}u'_{i}}$ ${\displaystyle u_{i}\in U_{i}}$ ${\displaystyle i\in I_{e}}$ ${\displaystyle \textstyle u_{i}=u'_{i}}$ ${\displaystyle i\in I_{e}}$ ${\displaystyle v\in V}$ ${\displaystyle \textstyle U_{i}}$

If one of these statements applies, the inner direct sum is called the . This direct sum becomes with ${\displaystyle V}$ ${\displaystyle U_{i}}$

${\displaystyle \textstyle \bigoplus _{i\in I}U_{i}}$

designated. The sub-module of is called a direct summand of , if there is a sub-module of with . The module is called directly indivisible or simply indivisible if it has no direct summand unequal . ${\displaystyle 0\neq U_{R}}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle V}$ ${\displaystyle M}$ ${\displaystyle U\oplus V=M}$ ${\displaystyle M}$ ${\displaystyle \{0\}}$

Examples

1. If there is a vector space over a field or a sloping body and a base of and is the subspace generated by for each , then is . ${\displaystyle V}$ ${\displaystyle \{x_{i}\mid i\in I\}}$ ${\displaystyle V}$ ${\displaystyle V_{i}}$ ${\displaystyle i\in I}$ ${\displaystyle x_{i}}$ ${\displaystyle \textstyle V=\bigoplus _{i\in I}V_{i}}$
2. Every simple module cannot be dismantled directly.
3. If there is an integrity ring and its quotient field , then as a module over is indivisible. ${\displaystyle R}$ ${\displaystyle K_{R}}$ ${\displaystyle K_{R}}$ ${\displaystyle R}$
4. ${\displaystyle 2\mathbb {Z} \subset \mathbb {Z} }$is not a direct summand, since there is no injective morphism are ${\displaystyle \mathbb {Z} /2\mathbb {Z} \to \mathbb {Z} }$

Special sub-modules

Maximum sub-modules

A sub-module is called maximum if no real sub-module of M actually contains it. ${\displaystyle U\subset M}$ ${\displaystyle U}$

${\displaystyle U\subset M}$is a maximum sub-module if and only if the factor module is simple . Every real sub-module of a finitely generated module is in a maximal sub-module. That is, in particular, each ring has maximum ideals. But there are also modules that do not contain any maximum sub-modules. So has no maximum sub-modules. ${\displaystyle M/U}$ ${\displaystyle \mathbb {Q} }$

Large sub-modules

definition

For a sub-module of are equivalent: ${\displaystyle U}$ ${\displaystyle M}$

• For all sub-modules with is . ${\displaystyle V\subset M}$ ${\displaystyle U\cap V=\{0\}}$ ${\displaystyle V=\{0\}}$
• For each there is a with . ${\displaystyle 0\neq x\in M}$ ${\displaystyle r\in R}$ ${\displaystyle 0\neq x\cdot r\in U}$

If a sub-module fulfills one of the equivalent properties, then capital in . Sometimes this is abbreviated to. ${\displaystyle U\subset M}$ ${\displaystyle U}$ ${\displaystyle M}$ ${\displaystyle U\trianglelefteq M}$

Examples

• In the as module, each sub-module is large. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \neq \{0\}}$
• The last example can be generalized. If a torsion-free Abelian group, then a subgroup is large if and only if the factor group is a torsion module. ${\displaystyle F}$ ${\displaystyle U\subset F}$ ${\displaystyle F/U}$
• If a prime number and a natural number are greater than 1, then is large in each sub-module. ${\displaystyle p}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Z} /(p^{n}\mathbb {Z} )}$
• In a semi-simple module , only the module itself is large in itself. ${\displaystyle M}$

properties

• If is big in and a sub-module of with , then big is in . ${\displaystyle U}$ ${\displaystyle M}$ ${\displaystyle V}$ ${\displaystyle M}$ ${\displaystyle U\subset V\subset M}$ ${\displaystyle V}$ ${\displaystyle M}$
• Is big in and big in , so is big in . ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle U}$ ${\displaystyle W}$
• Is a filtering family of sub-modules of, and is great in each , so is great in . ${\displaystyle (V_{i})_{i\in I}}$ ${\displaystyle M}$ ${\displaystyle U}$ ${\displaystyle V_{i}}$ ${\displaystyle U}$ ${\displaystyle \cup _{i}V_{i}}$
• If there are two families of sub-modules of and if the sum is direct, then the following applies: If all are large in , then large in . ${\displaystyle (U_{i})_{i\in I},(A_{i})_{i\in I}}$ ${\displaystyle M}$ ${\displaystyle A_{i}}$ ${\displaystyle U_{i}}$ ${\displaystyle A_{i}}$ ${\displaystyle \oplus _{i}U_{i}}$ ${\displaystyle \oplus _{i}A_{i}}$
• A sub-module is called closed if it is not part of any real main module. For each sub-module there is a closed sub-module , so in is big . ${\displaystyle U\subset M}$ ${\displaystyle U\subset M}$ ${\displaystyle {\overline {U}}}$ ${\displaystyle U}$ ${\displaystyle {\overline {U}}}$
• If there are two sub-modules of with , then there is a super-module of , which is maximal with regard to the property . It's big in . It is an average complement of . An average complement is by no means clearly determined. ${\displaystyle A,U}$ ${\displaystyle M}$ ${\displaystyle A\cap U=\{0\}}$ ${\displaystyle H}$ ${\displaystyle U}$ ${\displaystyle A\cap H=\{0\}}$ ${\displaystyle A\oplus H}$ ${\displaystyle M}$ ${\displaystyle H}$ ${\displaystyle A}$
• If there is a sub-module of , there is an intersecting complement of . To is an intersection complement of such that is a sub-module of . It's big in and complete in . ${\displaystyle A}$ ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle A'}$ ${\displaystyle A}$ ${\displaystyle A'}$ ${\displaystyle A''}$ ${\displaystyle A'}$ ${\displaystyle A}$ ${\displaystyle A''}$ ${\displaystyle A}$ ${\displaystyle A''}$ ${\displaystyle A''}$ ${\displaystyle M}$

The base of a module

If there is a module, the average of all large sub-modules is equal to the sum of all simple sub-modules. This sub-module is called the base of . It is the largest semi-simple sub-module of . He is designated with . Is ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \mathrm {So} (M)}$

${\displaystyle f\colon M\rightarrow N}$

is a homomorphism between modules , then is a sub-module of . In particular, this means that the base a -submodule of is when the Endmorphismenring of is. The base of the ring as a right module is a two-sided ideal. Also is ${\displaystyle M,N}$ ${\displaystyle f(\mathrm {So} (M))}$ ${\displaystyle \mathrm {So} (N)}$ ${\displaystyle S}$ ${\displaystyle M}$ ${\displaystyle S}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle R}$

${\displaystyle \mathrm {So} (\mathrm {So} (M))=\mathrm {So} (M)}$

The base is a pre-radical. It can be exchanged for direct sums. That means: if a family of sub-modules, the sum of which is direct, then is ${\displaystyle (A_{i})_{i\in I}}$

${\displaystyle \mathrm {So} (\oplus _{i}A_{i})=\oplus _{i}(\mathrm {So} (A_{i}))}$.

Small sub-modules

A sub-module is called small in if the following applies to all sub-modules of : Is , so is . ${\displaystyle A\subset M}$ ${\displaystyle M}$ ${\displaystyle U}$ ${\displaystyle M}$ ${\displaystyle A+U=M}$ ${\displaystyle U=M}$

Examples

• ${\displaystyle \{0\}}$ is small in each upper module.
• In a free Abelian group, only the module is small. ${\displaystyle \{0\}}$
• In every finitely generated subgroup is small as a submodule. ${\displaystyle \mathbb {Q} _{\mathbb {Z} }}$ ${\displaystyle \mathbb {Z} }$

properties

• The finite sum of small sub-modules is small.
• If is a homomorphism and is small in , then small in . ${\displaystyle f\colon M\rightarrow N}$ ${\displaystyle A}$ ${\displaystyle M}$ ${\displaystyle f(A)}$ ${\displaystyle N}$
• A cyclic sub-module is not small in if and only if there is a maximum sub-module with . ${\displaystyle aR\subset M}$ ${\displaystyle M}$ ${\displaystyle U\subset M}$ ${\displaystyle a\notin U}$

The radical of a module

The sum of all small sub-modules of is equal to the average of all maximum sub-modules of . This sub-module is called radical of . He is designated with . ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \operatorname {Rad(M)} }$

Properties of the radical

• If a homomorphism is a sub-module of (see also Jacobson radical ). The radical is a sub-function of identity. In particular, it is a two-sided ideal. ${\displaystyle f\colon M\rightarrow N}$ ${\displaystyle f(\operatorname {Rad} (M))}$ ${\displaystyle \operatorname {Rad} (N)}$ ${\displaystyle \operatorname {Rad} (R_{R})}$
• ${\displaystyle \operatorname {Rad} (M/\operatorname {Rad} (M))=\{0\}}$. The smallest sub-module of with is . ${\displaystyle C}$ ${\displaystyle M}$ ${\displaystyle \operatorname {Rad} (M/C)=\{0\}}$ ${\displaystyle \operatorname {Rad} (M)}$
• The radical can be exchanged for direct sums. This means that if a family of modules, then: . ${\displaystyle (M_{i}\mid i\in I)}$ ${\displaystyle \operatorname {Rad} (\oplus _{i}M_{i})=\oplus _{i}\operatorname {Rad} (M_{i})}$
• ${\displaystyle M\cdot \operatorname {Rad} (R_{R})}$is a sub-module of . ${\displaystyle \operatorname {Rad} (M)}$
• Is finitely generated, then is small in . ${\displaystyle M}$ ${\displaystyle \operatorname {Rad} (M)}$ ${\displaystyle M}$
• If the ideal is finitely generated and the ideal is a sub-module of , then small in . This is the Nakayama lemma . ${\displaystyle M}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle \operatorname {Rad} (R_{R})}$ ${\displaystyle M\cdot {\mathfrak {a}}}$ ${\displaystyle M}$

Individual evidence

1. ^ Kasch: Modules and Rings , 2.3.11
2. ^ Kasch: modules and rings. P. 34.
3. ^ Frank W. Anderson, Kent R. Fuller: Rings and Categories of Modules (= Graduate Texts in Mathematics 13). 2nd edition. Springer, New York NY et al. 1992, ISBN 3-540-97845-3 , p. 72.

literature

• Friedrich Kasch: modules and rings. Teubner, Stuttgart 1977, ISBN 3-519-02211-7 .
• Robert Wisbauer: Fundamentals of module and ring theory. Reinhard Fischer, Munich 1988, ISBN 3-88927-044-1 .