View source for Semi-linear mapping

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

<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>

<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>

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.

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" />.

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====
<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>



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