In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
s, most importantly
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 considere ...
s in certain
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 sp ...
s.
[Jacobson (2009), p. 142 and 147] These conditions played an important role in the development of the structure theory of commutative rings in the works of
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
,
Emmy Noether
Amalie Emmy NoetherEmmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noethe ...
, and
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
.
The conditions themselves can be stated in an abstract form, so that they make sense for any
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler.
Definition
A
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
(poset) ''P'' is said to satisfy the ascending chain condition (ACC) if no infinite strictly ascending sequence
:
of elements of ''P'' exists.
Equivalently,
[Proof: first, a strictly increasing sequence cannot stabilize, obviously. Conversely, suppose there is an ascending sequence that does not stabilize; then clearly it contains a strictly increasing (necessarily infinite) subsequence. Notice the proof does not use the full force of the axiom of choice.] every weakly ascending sequence
:
of elements of ''P'' eventually stabilizes, meaning that there exists a positive integer ''n'' such that
:
Similarly, ''P'' is said to satisfy the descending chain condition (DCC) if there is no
infinite descending chain
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some Set (mathematics), set X, which satisfies the following for all a, b and c in X:
# a ...
of elements of ''P''.
Equivalently, every weakly descending sequence
:
of elements of ''P'' eventually stabilizes.
Comments
* Assuming the
axiom of dependent choice In mathematics, the axiom of dependent choice, denoted by \mathsf , is a weak form of the axiom of choice ( \mathsf ) that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores whi ...
, the descending chain condition on (possibly infinite) poset ''P'' is equivalent to ''P'' being
well-founded
In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s& ...
: every nonempty subset of ''P'' has a minimal element (also called the minimal condition or minimum condition). A
totally ordered set
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
that is well-founded is a
well-ordered set
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-ord ...
.
* Similarly, the ascending chain condition is equivalent to ''P'' being converse well-founded (again, assuming dependent choice): every nonempty subset of ''P'' has a maximal element (the maximal condition or maximum condition).
* Every finite poset satisfies both the ascending and descending chain conditions, and thus is both well-founded and converse well-founded.
Example
Consider the ring
:
of integers. Each ideal of
consists of all multiples of some number
. For example, the ideal
:
consists of all multiples of
. Let
:
be the ideal consisting of all multiples of
. The ideal
is contained inside the ideal
, since every multiple of
is also a multiple of
. In turn, the ideal
is contained in the ideal
, since every multiple of
is a multiple of
. However, at this point there is no larger ideal; we have "topped out" at
.
In general, if
are ideals of
such that
is contained in
,
is contained in
, and so on, then there is some
for which all
. That is, after some point all the ideals are equal to each other. Therefore, the ideals of
satisfy the ascending chain condition, where ideals are ordered by set inclusion. Hence
is a
Noetherian ring
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-Noether ...
.
See also
*
Artinian
*
Ascending chain condition for principal ideals In abstract algebra, the ascending chain condition can be applied to the posets of principal left, principal right, or principal two-sided ideals of a ring, partially ordered by inclusion. The ascending chain condition on principal ideals (abbrevi ...
*
Krull dimension
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 t ...
*
Maximal condition on congruences
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
*
Noetherian In mathematics, the adjective Noetherian is used to describe Category_theory#Categories.2C_objects.2C_and_morphisms, objects that satisfy an ascending chain condition, ascending or descending chain condition on certain kinds of subobjects, meaning t ...
Notes
References
*
Atiyah, M. F., and I. G. MacDonald, ''
Introduction to Commutative Algebra
''Introduction to Commutative Algebra'' is a well-known commutative algebra textbook written by Michael Atiyah and Ian G. Macdonald. It deals with elementary concepts of commutative algebra including localization, primary decomposition, integral ...
'', Perseus Books, 1969,
*
Michiel Hazewinkel
Michiel Hazewinkel (born 22 June 1943) is a Dutch mathematician, and Emeritus Professor of Mathematics at the Centre for Mathematics and Computer Science and the University of Amsterdam, particularly known for his 1978 book ''Formal groups and a ...
, Nadiya Gubareni, V. V. Kirichenko. ''Algebras, rings and modules''.
Kluwer Academic Publishers
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, 2004.
* John B. Fraleigh, Victor J. Katz. ''A first course in abstract algebra''. Addison-Wesley Publishing Company. 5 ed., 1967.
*
Nathan Jacobson
Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician.
Biography
Born Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. He graduated from the University of Alabama in 1930 and was awar ...
. Basic Algebra I. Dover, 2009.
External links
*
{{DEFAULTSORT:Ascending Chain Condition
Commutative algebra
Order theory
Wellfoundedness