ringsrings

Ringsrings is a term used in abstract algebra to denote a recursive family of rings obtained by iterating standard ring-construction operations starting from a base ring. The name emphasizes the nested, self-similar nature of the construction, in which new rings are built from earlier ones by uniform, rule-based extensions.

Definitions and construction. Let R be a unital ring. A ringsrings over R is a finite sequence (R_0, R_1, ..., R_n) with R_0 = R and, for each k < n, R_{k+1} is obtained from R_k by one of two canonical extensions: forming the polynomial ring R_k[x], or forming the formal power series ring R_k[[x]]. Each step is accompanied by a canonical injective homomorphism R_k → R_{k+1}. The collection of all such finite sequences, modulo isomorphism, forms a tree-like object that is called the ringsrings over R.

Properties and examples. Every ring in a ringsrings sequence is a ring, and if the base ring R is commutative, all subsequent rings in the sequence are commutative as well. The construction typically increases algebraic invariants such as Krull dimension and can alter homological properties. From Z one obtains Z[x], Z[[x]], Z[x][y], Z[[x]][[t]], and so on, illustrating the breadth of possible ends.

Relation to other concepts. Ringsrings generalizes the familiar idea of repeatedly adjoining variables and completing rings, and it relates to pro- and ind- constructions in category theory when one considers infinite sequences. It is used in hypothetical discussions of recursive ring systems and in model-theoretic contexts to study how algebraic properties propagate through extensions.

See also: ring theory, polynomial ring, power series ring, pro-ring, ind-object.