Module homomorphism

In mathematics, a module homomorphism is a mapping between two modules and over a ring , which is compatible with the module structure. For example, it translates the addition of into the addition of . An addition can be translated twice. ${\displaystyle f\colon M\rightarrow N}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle N}$

1. First you add in and then translate with . ${\displaystyle M}$ ${\displaystyle f}$
2. You translate with the summands and calculate the sum in . ${\displaystyle f}$ ${\displaystyle N}$

With a homomorphism, the result is always the same. If the body is replaced by a ring in the definition of the linear mapping between vector spaces , a module homomorphism is obtained. The ring does not need to be commutative.

Homomorphism

definition

There are two right modules over a ring . A mapping is called homomorphism from to if the following applies to all and all : ${\displaystyle M,N}$ ${\displaystyle R}$ ${\displaystyle f\colon M\rightarrow N}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle m_{1},m_{2}\in M}$ ${\displaystyle r\in R}$

${\displaystyle f(m_{1}+m_{2})=f(m_{1})+f(m_{2})}$ and ${\displaystyle f(m_{1}\cdot r)=f(m_{1})\cdot r}$

The term homomorphism between left modules is explained accordingly: A mapping between two left modules and over the ring is called homomorphism from to , if the following applies to all and all : ${\displaystyle f\colon M\rightarrow N}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle m_{1},m_{2}\in M}$ ${\displaystyle r\in R}$

${\displaystyle f(m_{1}+m_{2})=f(m_{1})+f(m_{2})}$ and ${\displaystyle f(r\cdot m_{1})=r\cdot f(m_{1})}$

The set of homomorphisms from to is denoted by. ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle \operatorname {Hom} _{R}(M,N)}$

A homomorphism of a module in itself is called an endomorphism of . ${\displaystyle f\colon M\rightarrow M}$ ${\displaystyle M}$ ${\displaystyle M}$

If and two - - bimodules over rings and , then a mapping is called a homomorphism of SR bimodules if the following applies to all : ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle f\colon M\to N}$ ${\displaystyle m_{1},m_{2}\in M,\ s\in S,\ r\in R}$

${\displaystyle f(m_{1}+m_{2})=f(m_{1})+f(m_{2})}$and . ${\displaystyle f(s\cdot m\cdot r)=s\cdot f(m\cdot r)=s\cdot f(m)\cdot r}$

Examples

1. If there is any module, there is exactly one homomorphism , namely . It is an initial item in the legal module category. Likewise, there is only one homomorphism , the zero mapping ( for all ). It is also an end object. To sum up, when you say is a zero object. ${\displaystyle M}$ ${\displaystyle 0\colon \{0\}\rightarrow M}$ ${\displaystyle 0\mapsto 0\in M}$ ${\displaystyle \{0\}}$ ${\displaystyle M\to \{0\}}$ ${\displaystyle m\mapsto 0}$ ${\displaystyle m\in M}$ ${\displaystyle \{0\}}$ ${\displaystyle \{0\}}$
2. The identity is a homomorphism. ${\displaystyle \mathbf {1} _{M}\colon M\to M,\,m\mapsto m}$
3. The center of a ring is the set is a sub-ring of the ring . Is in the center of the ring, there is a homomorphism. ${\displaystyle R}$ ${\displaystyle Z=\{s\in R\mid s\cdot r=r\cdot s{\text{ für alle }}r\in R\}}$ ${\displaystyle R}$ ${\displaystyle s}$ ${\displaystyle l(s)\colon M\to M,\,m\mapsto m\cdot s}$
4. If there are two homomorphisms, their sum is a homomorphism. ${\displaystyle f,g\colon M\rightarrow N}$ ${\displaystyle f+g\colon M\to N,\,m\mapsto f(m)+g(m)}$

properties

• Is a homomorphism and is a sub-module of so is a sub-module of . In particular, is a sub-module of . This sub-module is called the core of homomorphism . It is often referred to as or simply . ${\displaystyle f\colon M\rightarrow N}$ ${\displaystyle V\hookrightarrow N}$ ${\displaystyle N}$ ${\displaystyle f^{-1}(V):=\{m\in M\mid f(m)\in V\}}$ ${\displaystyle M}$ ${\displaystyle f^{-1}(\{0\})}$ ${\displaystyle M}$ ${\displaystyle f}$ ${\displaystyle \operatorname {Kern} (f)}$ ${\displaystyle \operatorname {Ke} (f)}$
• If a sub-module of and is a module homomorphism, then a sub-module of . It's called picture from below . In particular , the image set of is a sub-module of . It is often referred to as or simply . ${\displaystyle U}$ ${\displaystyle M}$ ${\displaystyle f\colon M\to N}$ ${\displaystyle f(U)=\{f(u)\mid u\in U\}}$ ${\displaystyle N}$ ${\displaystyle U}$ ${\displaystyle f}$ ${\displaystyle f(M)}$ ${\displaystyle f}$ ${\displaystyle N}$ ${\displaystyle \operatorname {Bild} (f)}$ ${\displaystyle \operatorname {Bi} (f)}$
• The concatenation or composition of two homomorphisms is a homomorphism. The set of modules over a ring together with the homomorphisms form a category .
• If there is a module, the set of endomorphisms forms a unitary ring. The addition is the addition of the endomorphisms and the multiplication is the concatenation. ${\displaystyle M}$

Monomorphism

sentence

The following statements are equivalent for a homomorphism between modules. ${\displaystyle f\colon M_{R}\rightarrow N_{R}}$

1. ${\displaystyle \operatorname {Kern} (f)=\{m\in M\mid f(m)=0\}=\{0\}}$
2. For everyone with is . ${\displaystyle a,b\in M}$ ${\displaystyle f(a)=f(b)}$ ${\displaystyle a=b}$
3. ${\displaystyle f}$can be shortened on the left. This means that for all modules and all homomorphisms applies: . ${\displaystyle L_{R}}$ ${\displaystyle g,h:L_{R}\rightarrow M_{R}}$ ${\displaystyle fg=fh\Rightarrow g=h}$

If a homomorphism fulfills one and therefore all equivalent properties of the proposition, then it is called monomorphism between the modules. The third statement of the theorem says that in the sense of category theory there is a monomorphism . ${\displaystyle f\colon M_{R}\rightarrow N_{R}}$ ${\displaystyle f\,}$ ${\displaystyle f}$

Examples

1. If a sub-module , the inclusion map is a monomorphism. ${\displaystyle U\hookrightarrow M}$ ${\displaystyle \iota \colon U\to M,\,u\mapsto u}$
2. Every homomorphism of the set of integers into the rational numbers that is not the zero mapping is a monomorphism. ${\displaystyle \mathbb {Z} }$

Remarks

1. If and are monomorphisms, then there is a monomorphism. ${\displaystyle f\colon M\rightarrow N}$ ${\displaystyle g\colon N\rightarrow O}$ ${\displaystyle g\circ f}$
2. If there is a monomorphism, then it is a monomorphism. ${\displaystyle g\circ f}$ ${\displaystyle f}$
3. If there is a monomorphism, then it is . ${\displaystyle g\circ f}$ ${\displaystyle f(M)\cap \operatorname {Kern} (g)=\{0\}}$

Epimorphism

definition

The following statements are equivalent for a module homomorphism : ${\displaystyle f\in \operatorname {Hom} _{R}(M,N)}$

1. ${\displaystyle N/f(M)=0}$. Here, the factor module of N modulo f (M) . ${\displaystyle N/f(M)}$
2. The mapping is surjective . ${\displaystyle f}$
3. ${\displaystyle f}$can be shortened on the right. That is, for all modules and all homomorphisms applies: . ${\displaystyle P}$ ${\displaystyle g,h\in \operatorname {Hom} _{R}(N,P)}$ ${\displaystyle gf=hf\Rightarrow g=h}$

A homomorphism that fulfills one and therefore all of these properties is called an epimorphism. The third property of the theorem says that homomorphism in the sense of category theory is an epimorphism .

Examples

1. Identity is an epimorphism. ${\displaystyle M\to M,\,m\mapsto m}$
2. If there is an integrity ring and its quotient field , then every homomorphism is a monomorphism and an epimorphism. ${\displaystyle R}$ ${\displaystyle K}$ ${\displaystyle 0\neq f\in \operatorname {Hom} _{R}(K,K)}$
3. Let p be a prime number and be the smallest subring of the rational numbers that contains. If , then every endomorphism of that is not equal to the null map is an epimorphism. But multiplication by p is not a monomorphism. ${\displaystyle \mathbb {Z} [p^{-1}]=\{z\cdot p^{-i}\mid z\in \mathbb {Z} ,i\in \mathbb {N} \}}$ ${\displaystyle p^{-1}}$ ${\displaystyle M_{\mathbb {Z} }:=\mathbb {Z} [p^{-1}]/\mathbb {Z} }$ ${\displaystyle M_{\mathbb {Z} }}$

properties

1. The concatenation of epimorphisms is an epimorphism.
2. If and is an epimorphism, so is an epimorphism and it is . ${\displaystyle f\in \operatorname {Hom} _{R}(M,N),\quad g\in \operatorname {Hom} _{R}(N,P)}$ ${\displaystyle gf}$ ${\displaystyle g}$ ${\displaystyle \operatorname {Kern} (g)+f(M)=N}$

Isomorphisms

A homomorphism is called isomorphism if there is a homomorphism such that and is. This is precisely the case when there is a monomorphism and an epimorphism. It is the identity on the module and analog identity on the module . Two modules are called isomorphic, in signs if there is an isomorphism . ${\displaystyle f\colon M\rightarrow N}$ ${\displaystyle g\colon N\rightarrow M}$ ${\displaystyle g\circ f=\mathbf {1} _{M}}$ ${\displaystyle f\circ g=\mathbf {1} _{N}}$ ${\displaystyle f}$ ${\displaystyle \mathbf {1} _{M}}$ ${\displaystyle M}$ ${\displaystyle \mathbf {1} _{N}}$ ${\displaystyle N}$ ${\displaystyle M,N}$ ${\displaystyle M\cong N}$ ${\displaystyle \alpha :M\rightarrow N}$

Product decomposition of homomorphisms

Homomorphism theorem

If there is a homomorphism, then there is a clearly determined homomorphism such that it holds. It is with the canonical epimorphism. is always a monomorphism. If there is an epimorphism, then it is an isomorphism. ${\displaystyle f\colon M\rightarrow N}$ ${\displaystyle f^{*}\colon M/\operatorname {Kern} (f)\to N}$ ${\displaystyle f^{*}\pi =f}$ ${\displaystyle \pi \colon M\to M/\operatorname {Kern} (f)}$ ${\displaystyle m\mapsto m+\operatorname {Kern} (f)}$ ${\displaystyle f^{*}}$ ${\displaystyle f}$ ${\displaystyle f^{*}}$

The theorem of homomorphism says that the following diagram commutes .

${\displaystyle {\begin{array}{ccl}M&{\stackrel {f}{\longrightarrow }}&N\\\pi {\Big \downarrow }&&\parallel \mathbf {1} _{N}\\M/\operatorname {Kern} (f)&{\stackrel {f^{*}}{\longrightarrow }}&N\end{array}}}$

1. Isomorphism theorem

Be submodules of Then: . The isomorphism is ${\displaystyle U,V\hookrightarrow M}$ ${\displaystyle M}$ ${\displaystyle (U+V)/V\cong U/(U\cap V)}$ ${\displaystyle (U+V)/V\ni u+V\mapsto u+U\cap V\in U/(U\cap V)}$

Conclusion: be and sub-modules of with , so is . ${\displaystyle U\hookrightarrow M}$ ${\displaystyle V\hookrightarrow M}$ ${\displaystyle M}$ ${\displaystyle U\oplus V=M}$ ${\displaystyle U\cong M/V}$

2. Isomorphism Theorem

Let it be sub-modules of . Then: ${\displaystyle U\hookrightarrow V\hookrightarrow M}$ ${\displaystyle M}$

${\displaystyle M/V\cong (M/U)/(V/U)}$.

The hom functor

If modules are the set of homomorphisms . ${\displaystyle M,N}$ ${\displaystyle \operatorname {Hom} (M,N)}$ ${\displaystyle f\colon M\rightarrow N}$

Module properties of Hom

• The amount is an abelian group, if for two homomorphisms the sum is defined as follows: . ${\displaystyle \operatorname {Hom} (M,N)}$ ${\displaystyle f,g\colon M\rightarrow N}$ ${\displaystyle f+g\colon M\ni m\mapsto f(m)+g(m)\in N}$
• Is a bimodule, on the left side of a module on the ring and on the right side of a module on the ring , it is on the right side to form a module over the ring when looking for and defined . If the endomorphism ring is in particular , there is a module above the ring on the right-hand side . ${\displaystyle M}$ ${\displaystyle S-R}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle \operatorname {Hom} _{R}(M,N)}$ ${\displaystyle S}$ ${\displaystyle f\in \operatorname {Hom} _{R}(M,N)}$ ${\displaystyle s\in S}$ ${\displaystyle f\cdot s\colon M\ni m\mapsto f(s\cdot m)\in N}$ ${\displaystyle S}$ ${\displaystyle M}$ ${\displaystyle \operatorname {Hom} (M,N)}$ ${\displaystyle S}$
• Is a bimodule, on the left side of a module on the ring and on the right side over the ring , it is on the left side to the module over the ring when looking for and defined . ${\displaystyle N}$ ${\displaystyle S-R}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle \operatorname {Hom} _{R}(M,N)}$ ${\displaystyle S}$ ${\displaystyle f\in \operatorname {Hom} _{R}(M,N)}$ ${\displaystyle s\in S}$ ${\displaystyle s\cdot f\colon M\ni m\mapsto s\cdot f(m)\in N}$

The covariant functor Hom

If there is a module, the Abelian group is assigned to each module . The homomorphism is assigned to each homomorphism . It applies to all : . In addition, the identities are mapped to the corresponding identities. is a covariant functor from the category of modules above the ring to the category of Abelian groups. If a bimodule is a bimodule as above , then a functor is transferred from the module category to the module category . ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle \operatorname {Hom} (M,N)}$ ${\displaystyle \alpha \colon A\rightarrow B}$ ${\displaystyle \operatorname {Hom} (M,\alpha )\colon \operatorname {Hom} (M,A)\ni f\mapsto \alpha \circ f\in \operatorname {Hom} (M,B)}$ ${\displaystyle \alpha \colon A\rightarrow B\,\,,\beta \colon B\rightarrow C}$ ${\displaystyle \operatorname {Hom} (M,\beta \circ \alpha )=\operatorname {Hom} (M,\alpha )\circ \operatorname {Hom} (M,\beta )}$ ${\displaystyle \operatorname {Hom} (M,-)}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle S-R}$ ${\displaystyle \operatorname {Hom} (M,-)}$ ${\displaystyle R}$ ${\displaystyle S}$

Left Exactness of Hom

For a complex , that is, it holds , the following statements are equivalent: ${\displaystyle 0\rightarrow A\;{\overset {\alpha }{\rightarrow }}\;B\;{\overset {\beta }{\rightarrow }}\;C}$ ${\displaystyle \beta \circ \alpha =0}$

• ${\displaystyle 0\rightarrow A\;{\overset {\alpha }{\rightarrow }}\;B\;{\overset {\beta }{\rightarrow }}\;C}$ is exact.
• For all modules is exact . ${\displaystyle M}$ ${\displaystyle 0\rightarrow \operatorname {Hom} (M,A)\rightarrow \operatorname {Hom} (M,B)\rightarrow \operatorname {Hom} (M,C)}$
• There is a generator so that the sequence is exact. ${\displaystyle G}$ ${\displaystyle 0\rightarrow \operatorname {Hom} (G,A)\rightarrow \operatorname {Hom} (G,B)\rightarrow \operatorname {Hom} (G,C)}$

Even if is surjective, this is generally not the case for, that is, the Hom functor is generally not exact. The deviation from the exactness is measured by the Ext functor . ${\displaystyle \beta }$ ${\displaystyle \operatorname {Hom} (M,B)\rightarrow \operatorname {Hom} (M,C)}$

Individual evidence

1. ^ Friedrich Kasch, Modules and Rings , Teubner, Stuttgart 1977, page 57
2. ↑ Friedrich Kasch, Modules and Rings , Teubner, Stuttgart 1977, page 58

literature

• Frank W. Anderson and Kent R. Fuller: Rings and Categories of Modules . Springer, New-York 1992, ISBN 0-387-97845-3
• 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