Semi-linear mapping
A mapping from a (left) module
into a (left) module
over the same ring
, satisfying the conditions
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where ,
and
is some automorphism of
. One says that
is semi-linear relative to the automorphism
. A semi-linear mapping of vector spaces over the field
relative to complex conjugation
is also known as an anti-linear mapping. A semi-linear mapping of an
-module
into itself is known as a semi-linear transformation.
Example. A homothety of an -module
, i.e. a mapping
(
), where
is a fixed invertible element of
, is a semi-linear mapping relative to the automorphism
.
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
[1] | N. Bourbaki, "Algebra" , Elements of mathematics , 1 , Addison-Wesley (1973) pp. Chapts. I-III (Translated from French) |
Comments
A semi-linear transformation, i.e. a semi-linear mapping of a module into itself, is also called a semi-linear endomorphism.
Semi-linear mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-linear_mapping&oldid=30646