In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings. [1] [2] [3] These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether, and Emil Artin. The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler.
A partially ordered set (poset) P is said to satisfy the ascending chain condition (ACC) if no infinite strictly ascending sequence
of elements of P exists. [4] Equivalently, [a] 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 strictly descending chain of elements of P. [4] Equivalently, every weakly descending sequence
of elements of P eventually stabilizes.
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, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal.
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal. Some authors such as Bourbaki refer to PIDs as principal rings.
In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group.
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
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.
In mathematics, specifically ring theory, a principal ideal is an ideal in a ring that is generated by a single element of through multiplication by every element of The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset generated by a single element which is to say the set of all elements less than or equal to in
In mathematics, an algebra over a field is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".
In mathematics, the annihilator of a subset S of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by each element of S.
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules over a ring, keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.
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, specifically abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself. Both concepts are named for Emil Artin.
In algebra, the length of a module over a ring is a generalization of the dimension of a vector space which measures its size. page 153 It is defined to be the length of the longest chain of submodules. For vector spaces, the length equals the dimension. If is an algebra over a field , the length of a module is at most its dimension as a -vector space.
In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. The Noetherian property of a topological space can also be seen as a strong compactness condition, namely that every open subset of such a space is compact, and in fact it is equivalent to the seemingly stronger statement that every subset is compact.
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 is satisfied if there is no infinite strictly ascending chain of principal ideals of the given type (left/right/two-sided) in the ring, or said another way, every ascending chain is eventually constant.
In abstract algebra, a uniserial moduleM is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N1 and N2 of M, either or . A module is called a serial module if it is a direct sum of uniserial modules. A ring R is called a right uniserial ring if it is uniserial as a right module over itself, and likewise called a right serial ring if it is a right serial module over itself. Left uniserial and left serial rings are defined in a similar way, and are in general distinct from their right-sided counterparts.
In mathematics, the term "graded" has a number of meanings, mostly related: