# Semi-linear mapping

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A mapping from a (left) module into a (left) module over the same ring , satisfying the conditions  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.

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