Concept

I-adic topology

Summary
In commutative algebra, the mathematical study of commutative rings, adic topologies are a family of topologies on the underlying set of a module, generalizing the p-adic topologies on the integers. Let R be a commutative ring and M an R-module. Then each ideal a of R determines a topology on M called the a-adic topology, characterized by the pseudometric The family is a basis for this topology. With respect to the topology, the module operations of addition and scalar multiplication are continuous, so that M becomes a topological module. However, M need not be Hausdorff; it is Hausdorff if and only ifso that d becomes a genuine metric. Related to the usual terminology in topology, where a Hausdorff space is also called separated, in that case, the a-adic topology is called separated. By Krull's intersection theorem, if R is a Noetherian ring which is an integral domain or a local ring, it holds that for any proper ideal a of R. Thus under these conditions, for any proper ideal a of R and any R-module M, the a-adic topology on M is separated. For a submodule N of M, the canonical homomorphism to M/N induces a quotient topology which coincides with the a-adic topology. The analogous result is not necessarily true for the submodule N itself: the subspace topology need not be the a-adic topology. However, the two topologies coincide when R is Noetherian and M finitely generated. This follows from the Artin-Rees lemma. Completion (algebra) When M is Hausdorff, M can be completed as a metric space; the resulting space is denoted by and has the module structure obtained by extending the module operations by continuity. It is also the same as (or canonically isomorphic to): where the right-hand side is an inverse limit of quotient modules under natural projection. For example, let be a polynomial ring over a field k and a = (x1, ..., xn) the (unique) homogeneous maximal ideal. Then , the formal power series ring over k in n variables. As a consequence of the above, the a-adic closure of a submodule is This closure coincides with N whenever R is a-adically complete and M is finitely generated.
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Ontological neighbourhood
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Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers ; and p-adic integers. Commutative algebra is the main technical tool in the local study of schemes.
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