Basering

Basering, or base ring in English, is a term used in mathematics to denote the ring that supplies the scalars for another algebraic object. The basering serves as the coefficient ring with respect to which structures are defined, such as modules, algebras, or schemes.

In module theory, an R-module is a generalization of a vector space where scalars come from the ring R. Here R is the base ring. In algebra, an R-algebra is a ring A together with a homomorphism from R to the center of A, making A into an algebra over R; the ring R remains the base ring.

In algebraic geometry and related areas, many constructions are performed relative to a base ring. For example, a scheme is often described over a base ring R, and changing the base ring via a ring homomorphism R → S yields a base change, producing new objects like the spectrum of S and extension of scalars for modules.

Common examples of base rings include the ring of integers Z, any field k, and polynomial rings R[x] where R is a base ring. The choice of base ring influences properties such as torsion in modules and flatness under extension of scalars.

Related concepts include base change, extension of scalars, and the notion of a base field when the base ring is a field. Basering is a foundational concept in many areas of algebra and geometry.