commutative algebra 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. Prom ...

, the mathematical study of

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...

s, adic topologies are a family of topologies on elements of a

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...

, generalizing the -adic topologies on the

integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

Definition

Let be a

and an -

. Then each

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...

of determines a topology on called the -adic topology, characterized by the pseudometric

d(x,y)=2^\text

The family

\

is a basis for this topology.

Properties

With respect to the topology, the module operations of

addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...

and

multiplication Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...

are

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

, so that becomes a

topological module In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous. Examples A topological vector space is a topological module over a topological field. An abelian topological ...

. However, need not be Hausdorff; it is Hausdorff when and only when

\bigcap_ = 0\text

so that becomes a genuine

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...

. In that case, the -adic topology is called separated. The

canonical The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical examp ...

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

to induces a

quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...

which coincides with the -adic topology. The analogous result is not necessarily true for the submodule : the

subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...

need not be -adic. However, the two topologies coincide when is

Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...

and finitely generated, a result known as the Artin-Rees lemma.

Completion

When is Hausdorff, can be completed as a metric space; the resulting space is denoted by

\widehat M

and has the module structure obtained by extending the module operations by continuity. It is also the same as (or canonically isomorphic to):

\widehat = \varprojlim M/\mathfrak^n M

where the right-hand side is an

inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can ...

of quotient modules under natural projection. For example, let be a polynomial ring over a field and the maximal ideal. Then is a

formal power series ring In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...

Closed submodules

As a consequence of the above, the -adic closure of a submodule is

\overline=\bigcap_\text

This closure coincides with whenever is -adically complete and is finitely generated. is called

Zariski , birth_date = , birth_place = Kobrin, Russian Empire , death_date = , death_place = Brookline, Massachusetts, United States , nationality = American , field = Mathematics , work_institutions = ...

with respect to if every ideal in is -adically closed. There is a characterization: : is Zariski with respect to if and only if is contained in the

Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition y ...

of . In particular a Noetherian

local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebrai ...

is Zariski with respect to the

maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...

., exercise 6.

References

Sources

* * {{cite book, first=M. F., last=Atiyah, author-link1=Michael Atiyah, first2=I. G., last2=MacDonald, publisher=Addison-Wesley, location=Reading, MA, year=1969, title=Introduction to Commutative Algebra category:Commutative algebra category:Topology