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Semi-linear mapping

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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=30646
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article