Chain Of Ideals Research Articles

On the Baer ideal in algebras satisfying Capelli identities

The structure is investigated of the Baer ideal of a finitely generated algebra of arbitrary finite signature over an arbitrary field or over a Noetherian commutative-associative ring satisfying a system of Capelli identities of order n + 1. It is proved that the length of the Baer chain of ideals in such an algebra is at most n. It is proved that the quotient of this algebra modulo the largest nilpotent ideal is representable.

Lifting up an infinite chain of prime ideals to a valuation ring

We prove that for an arbitrary chain { P α } \{P_\alpha \} of prime ideals in an integral domain, there exists a valuation domain which has a chain of prime ideals { Q α } \{Q_\alpha \} lying over { P α } \{P_\alpha \} .

Properties of a standard form for a Boolean matrix

A standard form for a Boolean matrix is used to test a conjecture about the maximal length possible for chains of principal ideals in the semigroup of n × n Boolean matrices.

Meandering of trajectories of polynomial vector fields in the affine n-space

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in R n and an affine hyperplane. The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned by consecutive derivatives. This exposition constitutes an extended abstract of a forthcoming paper: only the basic steps are outlined here, with all technical details being either completely omitted or at best indicated.

Catenarity in quantum algebras

Various quantum algebras are shown to be catenary, i.e., all saturated chains of prime ideals between any two fixed primes have the same length. Further, Tauvel's formula relating the height of a prime ideal to the Gelfand-Kirillov dimension of the corresponding factor ring is established. These results are obtained for coordinate rings of quantum affine spaces, for quantized Weyl algebras, and for coordinate rings of complex quantum general linear groups, as well as for quantized enveloping algebras of maximal nilpotent subalgebras of semisimple complex Lie algebras.

Homomorphisms, ideals and commutativity in the Stone-Čech compactification of a discrete semigroup

In this paper we consider the Stone-Čech compactifications of discrete semigroups which are countable, commutative and cancellative. We show that, if S and T are two such semigroups, then any continuous injective homomorphism from T ∗ into S ∗ arises from a homomorphism mapping a cofinite subsemigroup of T into S. We also consider the structure of semiprincipal left ideals of βS, and show that any nonminimal ideal of this kind lies immediately above 2 c others for a certain order relationship on these ideals. We also show that it belongs to a reverse well-ordered chain of such ideals of type ω 1 ∗ , each maximal subject to being strictly less than all its predecessors in this ordering. We also show that each nonminimal idempotent of βS lies immediately above 2 c other idempotents for the order relation defined on idempotents by stating that α ⩽ β if α + β = β + α = α. Finally, if η is any nonminimal idempotent in βS, we show that the centre of the semigroup η + βS + η is contained in G + η, where G denotes the group generated by S.

Ideals and commutativity in β [formula omitted

In this paper, we consider the order relationship of semi-principal left ideals of β N . We prove that any nonminimal semi-principal left ideal belongs to a reverse well-ordered chain of such ideals of length ω 1, in which each ideal is maximal subject to being contained in all its predecessors. We also consider the order relationship defined on idempotents by stating that α⩽β if α+ β= β+ α= α, and prove that any nonminimal idempotent lies immediately above 2 c other idempotents. Finally, we show that for any nonminimal idempotent α in β N , Z +α is the centre of α+β N +α.

Chains of intermediate ideals in matrix near-rings

0. Introduction. It is well-known that there is a one to one correspondence between the set of(two-sided) ideals of a ring R with identity and the set of ideals of the corresponding matrix ring ~ (R), for any n e N. Moreover, the correspondence appears in a very trivial and natural way, as the reader would have seen in many a textbook. Since the introduction of matrices over arbi trary near-rings by Meldrum and van der Walt ([4]) in /984, ideals in the near-ring R and related ideals in the corresponding matrix near-ring JM~ (R) have been extensively studied. In 1987 van der Walt ([7]) has shown that there are in general more ideals in a matrix near-ring than in its base near-ring. He did this by relating ideals in the matrix near-ring to a given ideal in the base near-ring in two natural ways. These two methods coincide when the base near-ring is a ring, but not in the general near-ring setting, as an example in [7] shows. This was the first evidence of the preponderance of ideals in 1M~, (R) over those in R. Since then, more examples of this kind were constructed, even for near-rings which are "very nearly" rings. (See [t] and [3]). In a recent investigation by Meldrum and the author (see [2]), this process is t aken a step further: we have shown that, apart from the additional ideal in ]M, (R) that one can expect by performing the two natural constructions discussed in the previous paragraph on an ideal in R, there are, in general, even more ideals the so-called intermediate ideals lurking around in a matrix near-ring. One of the examples given in [2] was a zero-symmetric abelian near-ring and it was conjectured that arbitrarily long chains of intermediate ideals can be constructed by using this example. In this note it is shown that such chains of intermediate ideals do indeed exist.

Every Algebraic Chain Is the Congruence Lattice of a Ring

The equivalence of the following conditions on a chain L is proved: (1) L is algebraic; (2)There is a tight chain domain T (with identity) such that L is isomorphic to the chain of proper two-sided ideals of T and all two-sided ideals of T are idempotent; (3) L is isomorphic to the congruence lattice of a ring (not necessarily with identity).

C*-ALGEBRAS OF SINGULAR INTEGRAL OPERATORS AND TOEPLITZ OPERATORS ASSOCIATED WITH n-DIMENSIONAL FLOWS

For a given covariant representation π of an n-dimensional dynamical system (X, R n, α), we study the C*-algebras [Formula: see text] of abstract singular integral opertors and Toeplitz operators generated by π(C(X)) and Hilbert transforms corresponding to n linearly independent directions. Such an algebra has a chain of ideals [Formula: see text]. We compute all [Formula: see text] and show that for 0≤k≤n−1 each of these quotients is the direct sum of C*-algebras which can be thought of as [Formula: see text] for flows of lower dimensions. The ideal structure of [Formula: see text] is carefully studied. We also determine precisely when [Formula: see text] is a C*-algebra of type I.

Increasing Chains of Ideals and Orbit Closures in βℤ

Increasing Chains of Ideals and Orbit Closures in βℤ

Increasing chains of ideals and orbit closures in 𝛽𝑍

Given any discrete semigroup ( S , + ) (S, + ) , there is an extension of the operation to β S \beta S making ( β S , + ) (\beta S, + ) a left topological semigroup. The aim of this paper is, among other things, to prove that there exist strictly increasing chains of principle left ideals and of principal closed ideals in ( β Z , + ) (\beta \mathbb {Z}, + ) .

A maximal chain of principal ideals in the semigroup of binary relations on a finite set

A maximal chain of Fn+3−1 principal ideals in the semigroup of the binary relations on an n-element set X is constructed by representing a binary relation as a Boolean matrix. Here Fn stands for the n-th Fibonacci number.

Actions of divided power Hopf algebras

We study prime ideals in enveloping algebra smash products and use a duality construction to obtain results on prime ideals in rings on which divided power Hopf algebras act. These actions correspond to higher derivations. First, we consider chains of prime ideals in an enveloping algebra smash product over an arbitrary ring, where the Lie algebra is assumed to be finite dimensional abelian over a field of positive characteristic. We give a bound on the length of such a chain where the ideals all have the same intersection with the coefficient ring. Then using an explicit construction of a duality theorem of Blattner and Montgomery in this context, we are able to apply results on enveloping algebra smash products to study the invariant ideals of prime ideals in a ring, under a locally nilpotent divided power Hopf algebra action.

The polynomial ring over a Goldie ring need not be a Goldie ring

Rings satisfying chain conditions have been of interest for quite some time. The famous Wedderburn Theorem states that semisimple Artinian rings must be finite direct sums of matrix rings. Goldie’s theorem links rings with ascending chain conditions to rings with descending chain conditions, thus extending Wedderburn’s result. A ring R is said to be a (right) Goldie ring if R satisfies the ascending chain condition on (right) annihilator ideals and R contains no infinite direct sum of (right) ideals. From Goldie’s theorem we know that a semiprime (right) Goldie ring must be an order in a semisimple (right) Artinian ring. A natural question to arise is whether the matrix and polynomial rings over (right) Goldie rings are also necessarily (right) Goldie rings, Under certain additional hypotheses such as if the ring is an order in a (right) Artinian ring [4] or if the ring contains a certain type of uncountable set in its center [I], the answer is yes. Moreover, the second Goldie condition is always preserved [3]. Hence, as in the case for the matrix ring counterexample [2], we must explore the ascending chain condition on (right) annihilator ideals. In the next section we construct a commutative Goldie ring R whose polynomial ring R[t] contains two infinite sets of polynomials, {p,(t)} and (qj(t)}, such that pi(t) qj(t) = 0 iff i #j. This condition forces qk(t) to be in Ann((p,(t): i>k)) and qk(t) to be excluded from Ann({p,(t): i>k1)). Thus R[t] has an infinite ascending chain of annihilator ideals

On nil rings satisfying minimum condition on principal right ideals

In what follows under a ring we always mean an associative one. The main object of this paper is to describe the structure of nil rings with minimum condition on principal right ideals (Theorem 1). Let Z be the ring of integers. The minimum condition for principal right ideals means that every descending chain of principal right ideals (al),D(a2),~(a8),D ... D terminates after a finite number of steps where (a~),=aiZ+atA. By MHR-ring (MHL-ring) we mean a ring with minimum condition on principal right (left) ideals. MHR-rings were introduced by Sz~sz [7--9] and Faith [3]. Later Bass [2] considered perfect rings, i.e. MHR-rings with unit. Let us recall that a ring A is left T-nilpotent if for any sequence of elements a~ of A, i=1, 2, ..., n, ... there exists a positive integer n such that a~a~as...a,=O. For the basic notions and results of the radical theory we refer to Andrunakievi~--Rjabuhin [1], Szfisz [10], Wiegandt [11]. We denote by

Catenarity and Gelfand-Kirillov dimension in Ore extensions

If R is a commutative afline domain (for example, the coordinate ring of an irreducible algebraic variety), the sum ,of the height of any prime ideal in R and the dimension of the corresponding factor ring is the dimension of R. This implies that if Q and P are prime ideals of R with Q 2 P, any saturated chain of prime ideals from P to Q has length dim R/P dim R/Q, and so R is catenary. Schelter, Gabber, and others have proven analogous statements for some noncommutative affine rings R. In this paper we study the question of when an Ore extension of a commutative ring is catenary, giving both positive and negative results. We show that for locally finitedimensional Ore extensions of commutative affine rings, the above dimension statements are still true if we use the Gelfand-Kirillov dimension.

Auslander Algebras as Quasi-Hereditary Algebras

The notion of a quasi-hereditary algebra has been introduced by E. Cline, B. Parshall and L. Scott [7,2,5] in order to describe the so-called highest weight categories arising in the representation theory of Lie algebras and algebraic groups. Quasi-hereditary algebras are defined by the existence of a suitable chain of ideals, and the finite dimensional hereditary algebras are typical examples. In [3], also finite dimensional algebras of global dimension 2 are shown to be quasi-hereditary. Thus, the Auslander algebras are quasi-hereditary. Recall that the Auslander algebras A can be constructed in the following way. Let R be a representation-finite finite dimensional algebra; then A is the endomorphism algebra End(Affl), where M is a finite dimensional /{-module such that every indecomposable /{-module is isomorphic to a direct summand of M. We are going to introduce the notion of a splitting filtration on the class of all indecomposable /{-modules and show that in this way we obtain a heredity chain of ideals of A (see the definition below). Usually, there exist many splitting filtrations for a given R. Examples of splitting filtrations can be obtained from the Rojter measure, used by A. V. Rojter in his proof of the first Brauer-Thrall conjecture [6], or from the preprojective and preinjective partitions, introduced by M. Auslander and S. Smalo in [1]. Instead of dealing with finite dimensional algebras, we shall consider, more generally, semiprimary rings. Recall that an associative ring A with 1 is called semiprimary provided that its Jacobson radical TV is nilpotent and A/N is semisimple artinian. We say that an ideal J of A is a heredity ideal of A if P = J, JNJ = 0 and /, considered as a right ideal, is a projective ^-module. Following [2], a semiprimary ring A is said to be quasi-hereditary provided that there exists a chain

Rings graded by polycyclic-by-finite groups

We use the duality between group gradings and group actions to study polycyclic-by-finite group-graded rings. We show that, for such rings, graded Noetherian implies Noetherian and relate the graded Krull dimension to the Krull dimension. In addition we find a bound on the length of chains of prime ideals not containing homogeneous elements when the grading group is nilpotent-by-finite. These results have suitable corollaries for strongly group-graded rings. Our work extends several results on skew group rings, crossed products and group-graded rings.

Closed and prime ideals in the algebra of bounded analytic functions

Let H∞ be the Banach algebra of all bounded analytic functions in the unit disc. We present a complete description of the closed primary (respectively prime) ideals contained in a maximal ideal of the Shilov boundary of H∞. The paper is also concerned with chains of prime ideals in H∞.