Slinear

Slinear, or S-linear, is a term used in mathematics to denote linearity with respect to a specified subring S of the scalar field. An S-linear map is a function between S-modules that preserves addition and is homogeneous with respect to the action of S: f(x+y) = f(x) + f(y) and f(sx) = s f(x) for all s in S and x in the domain. Equivalently, such a map is an S-module homomorphism.

In module theory and linear algebra, S-linear maps generalize ordinary linear maps by allowing the scalar action to be restricted to a subring S. If V and W are vector spaces over a field F and S is a subfield of F, then every F-linear map is S-linear, but an S-linear map need not be F-linear. A standard example occurs when F = C and S = R: viewing C as an R-vector space, the complex conjugation map f(z) = z̄ is R-linear but not C-linear.

Usage of the term varies; S-linear is most commonly encountered in the language of module theory and category theory, where it is synonymous with S-module homomorphism. Outside mathematics, the string Slinear may be used as a product name or acronym, depending on the context. See also linear map, module, and ring.