Semi-linear mapping

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