In mathematics, a commutative ring R is catenary if for any pair of prime ideals p, q, any two strictly increasing chains
of prime ideals are contained in maximal strictly increasing chains from p to q of the same (finite) length. In a geometric situation, in which the dimension of an algebraic variety attached to a prime ideal will decrease as the prime ideal becomes bigger, the length of such a chain n is usually the difference in dimensions.
A ring is called universally catenary if all finitely generated algebras over it are catenary rings.
The word 'catenary' is derived from the Latin word catena, which means "chain".
There is the following chain of inclusions.
Suppose that A is a Noetherian domain and B is a domain containing A that is finitely generated over A. If P is a prime ideal of B and p its intersection with A, then
The dimension formula for universally catenary rings says that equality holds if A is universally catenary. Here κ(P) is the residue field of P and tr.deg. means the transcendence degree (of quotient fields). In fact, when A is not universally catenary, but , then equality also holds. [1]
Almost all Noetherian rings that appear in algebraic geometry are universally catenary. In particular the following rings are universally catenary:
It is delicate to construct examples of Noetherian rings that are not universally catenary. The first example was found by MasayoshiNagata ( 1956 , 1962 , page 203 example 2), who found a 2-dimensional Noetherian local domain that is catenary but not universally catenary.
Nagata's example is as follows. Choose a field k and a formal power series z=Σi>0aixi in the ring S of formal power series in x over k such that z and x are algebraically independent.
Define z1 = z and zi+1=zi/x–ai.
Let R be the (non-Noetherian) ring generated by x and all the elements zi.
Let m be the ideal (x), and let n be the ideal generated by x–1 and all the elements zi. These are both maximal ideals of R, with residue fields isomorphic to k. The local ring Rm is a regular local ring of dimension 1 (the proof of this uses the fact that z and x are algebraically independent) and the local ring Rn is a regular Noetherian local ring of dimension 2.
Let B be the localization of R with respect to all elements not in either m or n. Then B is a 2-dimensional Noetherian semi-local ring with 2 maximal ideals, mB (of height 1) and nB (of height 2).
Let I be the Jacobson radical of B, and let A = k+I. The ring A is a local domain of dimension 2 with maximal ideal I, so is catenary because all 2-dimensional local domains are catenary. The ring A is Noetherian because B is Noetherian and is a finite A-module. However A is not universally catenary, because if it were then the ideal mB of B would have the same height as mB∩A by the dimension formula for universally catenary rings, but the latter ideal has height equal to dim(A)=2.
Nagata's example is also a quasi-excellent ring, so gives an example of a quasi-excellent ring that is not an excellent ring.
In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence of left ideals has a largest element; that is, there exists an n such that:
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 specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.
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.
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.
In mathematics, specifically abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition.
In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways.
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of generators of m. Then by Krull's principal ideal theorem n ≥ dim A, and A is defined to be regular if n = dim A.
In commutative algebra, an element b of a commutative ring B is said to be integral over a subring A of B if b is a root of some monic polynomial over A.
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal.
In commutative algebra, an N-1 ring is an integral domain whose integral closure in its quotient field is a finitely generated -module. It is called a Japanese ring if for every finite extension of its quotient field , the integral closure of in is a finitely generated -module. A ring is called universally Japanese if every finitely generated integral domain over it is Japanese, and is called a Nagata ring, named for Masayoshi Nagata, or a pseudo-geometric ring if it is Noetherian and universally Japanese. A ring is called geometric if it is the local ring of an algebraic variety or a completion of such a local ring, but this concept is not used much.
In commutative algebra, a G-ring or Grothendieck ring is a Noetherian ring such that the map of any of its local rings to the completion is regular. Almost all Noetherian rings that occur naturally in algebraic geometry or number theory are G-rings, and it is quite hard to construct examples of Noetherian rings that are not G-rings. The concept is named after Alexander Grothendieck.
In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety. The need of a theory for such an apparently simple notion results from the existence of many definitions of dimension that are equivalent only in the most regular cases. A large part of dimension theory consists in studying the conditions under which several dimensions are equal, and many important classes of commutative rings may be defined as the rings such that two dimensions are equal; for example, a regular ring is a commutative ring such that the homological dimension is equal to the Krull dimension.
In commutative algebra, an integrally closed domainA is an integral domain whose integral closure in its field of fractions is A itself. Spelled out, this means that if x is an element of the field of fractions of A that is a root of a monic polynomial with coefficients in A, then x is itself an element of A. Many well-studied domains are integrally closed, as shown by the following chain of class inclusions:
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring.
In commutative algebra, the Krull–Akizuki theorem states the following: Let A be a one-dimensional reduced noetherian ring, K its total ring of fractions. Suppose L is a finite extension of K. If and B is reduced, then B is a one-dimensional noetherian ring. Furthermore, for every nonzero ideal of B, is finite over A.
This is a glossary of commutative algebra.
In algebra, an analytically unramified ring is a local ring whose completion is reduced.
In algebra, specifically in the theory of commutative rings, a quasi-unmixed ring is a Noetherian ring such that for each prime ideal p, the completion of the localization Ap is equidimensional, i.e. for each minimal prime ideal q in the completion , = the Krull dimension of Ap.
In abstract algebra, the Eakin–Nagata theorem states: given commutative rings such that is finitely generated as a module over , if is a Noetherian ring, then is a Noetherian ring.
