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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s0842101.png" /> from a (left) [[Module|module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s0842102.png" /> into a (left) module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s0842103.png" /> over the same ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s0842104.png" />, satisfying the conditions
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{{MSC|15}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s0842105.png" /></td> </tr></table>
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A ''semi-linear mapping'' is a mapping $\def\a{\alpha}\a$ from a (left)
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[[Module|module]] $M$ into a (left) module $N$ over the same ring $A$, satisfying the conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s0842106.png" /></td> </tr></table>
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$$\a(x+y)=\a(x)+\a(y),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s0842107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s0842108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s0842109.png" /> is some automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s08421010.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s08421011.png" /> is semi-linear relative to the automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s08421012.png" />. A semi-linear mapping of vector spaces over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s08421013.png" /> relative to complex conjugation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s08421014.png" /> is also known as an anti-linear mapping. A semi-linear mapping of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s08421015.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s08421016.png" /> into itself is known as a semi-linear transformation.
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$$\def\s{\sigma}\a(cx)=c^\s\a(x)$$
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where $x,y\in M$, $c\in A$ and $c\mapsto c^\s$ is some automorphism of $A$. One says that $\a$ is semi-linear relative to the automorphism $\s$. A semi-linear mapping of vector spaces over the field $\C$ relative to complex conjugation $c^\s = \bar c$ is also known as an anti-linear mapping. A semi-linear mapping of an $A$-module $M$ into itself is known as a semi-linear transformation.
  
Example. A homothety of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s08421018.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s08421019.png" />, i.e. a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s08421020.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s08421021.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s08421022.png" /> is a fixed invertible element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s08421023.png" />, is a semi-linear mapping relative to the automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084210/s08421024.png" />.
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Example. A homothety of an $A$-module $M$, i.e. a mapping $x\mapsto ax$ ($x\in M$), where $a$ is a fixed invertible element of $A$, is a semi-linear mapping relative to the automorphism $c^\s = aca^{-1}$.
  
 
Many properties of linear mappings and homomorphisms of modules remain valid for semi-linear mappings. In particular, the kernel and image of a semi-linear mapping are submodules; semi-linear mappings of free modules with finite bases are completely determined by their matrices; one can define the rank of a semi-linear mapping of vector spaces, which is equal to the rank of its matrix; etc.
 
Many properties of linear mappings and homomorphisms of modules remain valid for semi-linear mappings. In particular, the kernel and image of a semi-linear mapping are submodules; semi-linear mappings of free modules with finite bases are completely determined by their matrices; one can define the rank of a semi-linear mapping of vector spaces, which is equal to the rank of its matrix; etc.
  
====References====
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A semi-linear transformation, i.e., a semi-linear mapping of a module into itself, is also called a semi-linear endomorphism.
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Algebra" , ''Elements of mathematics'' , '''1''' , Addison-Wesley  (1973)  pp. Chapts. I-III  (Translated from French)</TD></TR></table>
 
  
  
 
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====References====
====Comments====
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{|
A semi-linear transformation, i.e. a semi-linear mapping of a module into itself, is also called a semi-linear endomorphism.
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|-
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|valign="top"|{{Ref|Bo}}||valign="top"|  N. Bourbaki,  "Algebra",
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''Elements of mathematics'', '''1''', Addison-Wesley  (1973)
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pp. Chapts. I-III  (Translated from French) 
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{{MR|0354207}} {{ZBL|1111.00001}}
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|-
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|}

Latest revision as of 22:23, 2 November 2013

2020 Mathematics Subject Classification: Primary: 15-XX [MSN][ZBL]

A semi-linear mapping is a mapping $\def\a{\alpha}\a$ from a (left) module $M$ into a (left) module $N$ over the same ring $A$, satisfying the conditions

$$\a(x+y)=\a(x)+\a(y),$$

$$\def\s{\sigma}\a(cx)=c^\s\a(x)$$ where $x,y\in M$, $c\in A$ and $c\mapsto c^\s$ is some automorphism of $A$. One says that $\a$ is semi-linear relative to the automorphism $\s$. A semi-linear mapping of vector spaces over the field $\C$ relative to complex conjugation $c^\s = \bar c$ is also known as an anti-linear mapping. A semi-linear mapping of an $A$-module $M$ into itself is known as a semi-linear transformation.

Example. A homothety of an $A$-module $M$, i.e. a mapping $x\mapsto ax$ ($x\in M$), where $a$ is a fixed invertible element of $A$, is a semi-linear mapping relative to the automorphism $c^\s = aca^{-1}$.

Many properties of linear mappings and homomorphisms of modules remain valid for semi-linear mappings. In particular, the kernel and image of a semi-linear mapping are submodules; semi-linear mappings of free modules with finite bases are completely determined by their matrices; one can define the rank of a semi-linear mapping of vector spaces, which is equal to the rank of its matrix; etc.

A semi-linear transformation, i.e., a semi-linear mapping of a module into itself, is also called a semi-linear endomorphism.


References

[Bo] N. Bourbaki, "Algebra",

Elements of mathematics, 1, Addison-Wesley (1973) pp. Chapts. I-III (Translated from French) MR0354207 Zbl 1111.00001

How to Cite This Entry:
Semi-linear mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-linear_mapping&oldid=18709
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article