Ideal theory

In mathematics, ideal theory is the theory of ideals in commutative rings; and is the precursor name for the contemporary subject of commutative algebra. The name grew out of the central considerations, such as the Lasker–Noether theorem in algebraic geometry, and the ideal class group in algebraic number theory, of the commutative algebra of the first quarter of the twentieth century. It was used in the influential van der Waerden text on abstract algebra from around 1930.
The ideal theory in question had been based on elimination theory, but in line with David Hilbert's taste moved away from algorithmic methods. Gröbner basis theory has now reversed the trend, for computer algebra.
The importance of the idea of a module, more general than an "ideal", probably led to the perception that "ideal theory" was too narrow a description. Valuation theory, too, was an important technical extension, and was used by Helmut Hasse and Oscar Zariski. Bourbaki used "commutative algebra"; sometimes "local algebra" is applied to the theory of local rings. Douglas Northcott's 1953 Cambridge Tract "Ideal Theory" (reissued 2004 under the same title) was one of the final appearances of the name.
Topology determined by an ideal.
Let "R" be a ring and "M" an "R"-module. Then each ideal formula_1 of "R" determines a topology on "M" called the formula_1-adic topology such that a subset "U" of "M" is open if and only if for each "x" in "U" there exists a positive integer "n" such that
With respect to this formula_1-adic topology, formula_5 is a basis of neighbourhoods of formula_6 and makes the module operations continuous; in particular, formula_7 is a possibly non-Hausdorff topological group. Also, "M" is a Hausdorff topological space if and only if formula_8 Moreover, when formula_7 is Hausdorff, the topology is the same as the metric space topology given by defining the distance function: formula_10 for formula_11, where formula_12 is an integer such that formula_13.
Given a submodule "N" of "M", the formula_1-closure of "N" in "M" is equal to formula_15, as shown easily.
Now, "a priori", on a submodule "N" of "M", there are two natural formula_1-topologies: the subspace topology induced by the formula_1-adic topology on "M" and the formula_1-adic topology on "N". However, when formula_19 is Noetherian and formula_7 is finite over it, those two topologies coincide as a consequence of the Artin–Rees lemma.
When formula_7 is Hausdorff, formula_7 can be completed as a metric space; the resulting space is denoted by formula_23 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 the completion of the module formula_7 with respect to formula_1.
Example: Let formula_27 be a polynomial ring over a field and formula_28 the maximal ideal. Then formula_29 is a formal power series ring.
"R" is called a Zariski ring with respect to formula_1 if every ideal in "R" is formula_1-closed. There is a characterization:
In particular a Noetherian local ring is a Zariski ring with respect to the maximal ideal.
System of parameters.
A system of parameters for a local Noetherian ring of Krull dimension "d" with maximal ideal "m" is a set of elements "x"1, ..., "x""d" that satisfies any of the following equivalent conditions:
Every local Noetherian ring admits a system of parameters.
It is not possible for fewer than "d" elements to generate an ideal whose radical is "m" because then the dimension of "R" would be less than "d".
If "M" is a "k"-dimensional module over a local ring, then "x"1, ..., "x""k" is a system of parameters for "M" if the length of is finite.
Reduction theory.
The reduction theory goes back to the influential 1954 paper by Northcott and Rees, the paper that introduced the basic notions. In algebraic geometry, the theory is among the essential tools to extract detailed information about the behaviors of blow-ups.
Given ideals "J" ⊂ "I" in a ring "R", the ideal "J" is said to be a "reduction" of "I" if there is some integer "m" > 0 such that formula_34. For such ideals, immediately from the definition, the following hold:
If "R" is a Noetherian ring, then "J" is a reduction of "I" if and only if the Rees algebra "R"["It"] is finite over "R"["Jt"]. (This is the reason for the relation to a blow up.)
A closely related notion is that of analytic spread. By definition, the fiber cone ring of a Noetherian local ring ("R", formula_36) along an ideal "I" is
The Krull dimension of formula_38 is called the "analytic spread" of "I". Given a reduction formula_39, the minimum number of generators of "J" is at least the analytic spread of "I". Also, a partial converse holds for infinite fields: if formula_40 is infinite and if the integer formula_41 is the analytic spread of "I", then each reduction of "I" contains a reduction generated by formula_41 elements.
Local cohomology in ideal theory.
Local cohomology can sometimes be used to obtain information on an ideal. This section assumes some familiarity with sheaf theory and scheme theory.
Let formula_7 be a module over a ring formula_19 and formula_45 an ideal. Then formula_7 determines the sheaf formula_47 on formula_48 (the restriction to "Y" of the sheaf associated to "M"). Unwinding the definition, one sees:
Here, formula_50 is called the ideal transform of formula_7 with respect to formula_45.