Ringfolgen

Ringfolgen are sequences of rings connected by ring homomorphisms, typically organized as a directed system of rings. In this context, one speaks of a collection of rings (R_i) indexed by a directed set I, together with maps φ_i^j: R_i → R_j for all i ≤ j, satisfying φ_i^i = id and φ_j^k ∘ φ_i^j = φ_i^k for all i ≤ j ≤ k. When I is the natural numbers with maps R_n → R_{n+1}, the structure is called a direct system; if the maps go from R_{n+1} to R_n, it is an inverse system.

From a Ringfolge one can form universal constructions known as limits. The direct limit (colimit) of a direct system is the most general ring receiving compatible maps from all R_i. The inverse limit (limit) of an inverse system is a ring consisting of compatible families of elements in the R_i, embedded in the product ∏ R_i with a coherence condition.

Common examples illustrate the concepts. A direct system can model localization processes: Z ⊆ Z[1/2] ⊆ Z[1/2, 1/3] ⊆ …, whose direct limit is the localization of Z at all primes, isomorphic to the field of rational numbers Q. An inverse system arises in the construction of p-adic integers: Z_p ≅ lim← Z/p^nZ with the natural reduction maps Z/p^{n+1}Z → Z/p^nZ.

Ringfolgen also appear in the study of filtrations and completions, where a chain of subrings R_0 ⊆ R_1 ⊆ R_2 ⊆ … yields associated graded structures or completion with respect to the filtration. In category-theoretic terms, a Ringfolge is a functor from a directed index category into the category of rings, and limits describe universal properties of these constructions.