Chain Conditions for Modular Lattices with Finite Group Actions

1979 ◽  
Vol 31 (3) ◽  
pp. 558-564 ◽  
Author(s):  
Joe W. Fisher
Keyword(s):  
Modular Lattice ◽  
Chain Condition ◽  
Common Fixed Points ◽  
Finite Subgroup ◽  
Modular Lattices ◽  
Finite Group Actions ◽  
Chain Conditions ◽  
Combinatorial Result ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

This paper establishes the following combinatorial result concerning the automorphisms of a modular lattice.THEOREM. Let M be a modular lattice and let G be a finite subgroup of the automorphism group of M. If the sublattice, MG, of (common) fixed points (under G) satisfies any of a large class of chain conditions, then M satisfies the same chain condition. Some chain conditions in this class are the following: the ascending chain condition; the descending chain condition; Krull dimension; the property of having no uncountable chains, no chains order-isomorphic to the rational numbers; etc.

Banach algebras satisfying certain chain conditions on closed ideals

2020 ◽  
Vol 57 (3) ◽  
pp. 290-297
Author(s):  
Abdullah Alahmari ◽  
Falih A. Aldosray ◽  
Mohamed Mabrouk
Keyword(s):  
Banach Algebra ◽  
Jacobson Radical ◽  
Banach Algebras ◽  
Chain Condition ◽  
Chain Conditions ◽  
Finite Dimensional ◽  
Descending Chain Condition ◽  
Unital Banach Algebra ◽  
Ascending Chain Condition

AbstractLet 𝔄 be a unital Banach algebra and ℜ its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra 𝔄 satisfies the descending chain condition on closed left ideals then 𝔄/ℜ is finite dimensional. We also prove that a C*-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.

Goldie Dimension and Chain Conditions for Modular Lattices with Finite Group Actions

1986 ◽  
Vol 29 (3) ◽  
pp. 274-280 ◽  
Author(s):  
Piotr Grzeszczuk ◽  
Edmund R. Puczyłowski
Keyword(s):  
Finite Group ◽  
Fixed Points ◽  
Modular Lattice ◽  
Group Actions ◽  
Modular Lattices ◽  
Finite Group Actions ◽  
Chain Conditions ◽  
Goldie Dimension ◽  
A Finite Group

AbstractA relation between Goldie dimensions of a modular lattice L and its sublattice LG of fixed points under a finite group G of automorphisms of L is obtained. The method used also gives a relation between ACC (DCC) for L and for LG. The results obtained are applied to rings and modules.

Varieties generated by 2-testable monoids

2012 ◽  
Vol 49 (3) ◽  
pp. 366-389 ◽  
Author(s):  
Edmond Lee
Keyword(s):  
Present Article ◽  
Chain Condition ◽  
Finite Basis ◽  
Finitely Based ◽  
Finitely Generated ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

The smallest monoid containing a 2-testable semigroup is defined to be a 2-testable monoid. The well-known Brandt monoid B21 of order six is an example of a 2-testable monoid. The finite basis problem for 2-testable monoids was recently addressed and solved. The present article continues with the investigation by describing all monoid varieties generated by 2-testable monoids. It is shown that there are 28 such varieties, all of which are finitely generated and precisely 19 of which are finitely based. As a comparison, the sub-variety lattice of the monoid variety generated by the monoid B21 is examined. This lattice has infinite width, satisfies neither the ascending chain condition nor the descending chain condition, and contains non-finitely generated varieties.

On Prime Rings with Ascending Chain Condition on Annihilator Right Ideals and Nonzero Infective Right Ideals

1971 ◽  
Vol 14 (3) ◽  
pp. 443-444 ◽  
Author(s):  
Kwangil Koh ◽  
A. C. Mewborn
Keyword(s):  
Prime Ring ◽  
Chain Condition ◽  
Prime Rings ◽  
Simple Ring ◽  
Descending Chain Condition ◽  
Image Position ◽  
The Right ◽  
Ascending Chain Condition

If I is a right ideal of a ring R, I is said to be an annihilator right ideal provided that there is a subset S in R such thatI is said to be injective if it is injective as a submodule of the right regular R-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective.

N-series and tame near-rings

1980 ◽  
Vol 86 (1-2) ◽  
pp. 153-163 ◽  
Author(s):  
C. G. Lyons ◽  
J.D.P. Meldrum
Keyword(s):  
Chain Condition ◽  
Chain Conditions ◽  
Descending Chain Condition

SynopsisLet N be a zero-symmetric near-ring with identity and let G be an N-group. We consider in this paper nilpotent ideals of N and N-series of G and we seek to link these two ideas by defining characterizing series for nilpotent ideals. These often exist and in most cases a minimal characterizing series exists. Another special N-series is a radical series, that is a shortest N-series with a maximal annihilator. These are linked to appropriate characterizing series. We apply these ideas to obtain characterizing series for the radical of a tame near-ring N, and to show that these exist if either G has both chain conditions on N-ideals or N has the descending chain condition on right ideals. In the latter case this provides a new proof of the nilpotency of the radical of a tame near-ring with DCCR, and an internal method for constructing minimal and maximal characterizing series for the radical.

scholarly journals Some conditions for finiteness of a ring

1988 ◽  
Vol 11 (2) ◽  
pp. 239-242 ◽  
Author(s):  
Howard E. Bell
Keyword(s):  
Chain Condition ◽  
Zero Divisors ◽  
Periodic Ring ◽  
Descending Chain Condition ◽  
Nil Ring ◽  
Ascending Chain Condition

Extending a result of Putcha and Yaqub, we prove that a non-nil ring must be finite if it has both ascending chain condition and descending chain condition on non-nil subrings. We also prove that a periodic ring with only finitely many non-central zero divisors must be either finite or commutative.

scholarly journals The descending chain condition in modular lattices

1972 ◽  
Vol 14 (4) ◽  
pp. 443-444
Author(s):  
Thomas G. Newman
Keyword(s):  
Elementary Proof ◽  
Chain Condition ◽  
Modular Lattices ◽  
Descending Chain Condition

In a recent paper Kovács [1] studied join-continuous modular lattices which satisfy the following conditions: (i) every element is a join of finitely many join-irredicibles, and, (ii) the set of join-irreducibles satisfies the descending chain condition. He was able to prove that such a lattice must itself satisfy the descending chain condition. Interest was expressed in whether or not one could obtain the same result without the assumption of modularity and/or of join-continuity. In this paper we give an elementary proof of this result without the assumption of join- continuity (which of course must then follow as a consequence of the descending chain condition). In addition we give a suitable example to show that modularity may not be omitted in general.

Nil Subrings of Goldie Rings are Nilpotent

1969 ◽  
Vol 21 ◽  
pp. 904-907 ◽  
Author(s):  
Charles Lanski
Keyword(s):  
Chain Condition ◽  
Chain Conditions ◽  
Direct Sum ◽  
Left Annihilator ◽  
Ascending Chain Condition

Herstein and Small have shown (1) that nil rings which satisfy certain chain conditions are nilpotent. In particular, this is true for nil (left) Goldie rings. The result obtained here is a generalization of their result to the case of any nil subring of a Goldie ring.Definition. Lis a left annihilator in the ring R if there exists a subset S ⊂ R with L = {x∈ R|xS= 0}. In this case we write L= l(S). A right annihilator K = r(S) is defined similarly.Definition. A ring R satisfies the ascending chain condition on left annihilators if any ascending chain of left annihilators terminates at some point. We recall the well-known fact that this condition is inherited by subrings.Definition. R is a Goldie ring if R has no infinite direct sum of left ideals and has the ascending chain condition on left annihilators.

scholarly journals COVERS FOR S-ACTS AND CONDITION (A) FOR A MONOID S

2014 ◽  
Vol 57 (2) ◽  
pp. 323-341
Author(s):  
ALEX BAILEY ◽  
VICTORIA GOULD ◽  
MIKLÓS HARTMANN ◽  
JAMES RENSHAW ◽  
LUBNA SHAHEEN
Keyword(s):  
Chain Condition ◽  
Projective Cover ◽  
Descending Chain Condition ◽  
Strongly Flat ◽  
Ascending Chain Condition

AbstractA monoid S satisfies Condition (A) if every locally cyclic left S-act is cyclic. This condition first arose in Isbell's work on left perfect monoids, that is, monoids such that every left S-act has a projective cover. Isbell showed that S is left perfect if and only if every cyclic left S-act has a projective cover and Condition (A) holds. Fountain built on Isbell's work to show that S is left perfect if and only if it satisfies Condition (A) together with the descending chain condition on principal right ideals, MR. We note that a ring is left perfect (with an analogous definition) if and only if it satisfies MR. The appearance of Condition (A) in this context is, therefore, monoid specific. Condition (A) has a number of alternative characterisations, in particular, it is equivalent to the ascending chain condition on cyclic subacts of any left S-act. In spite of this, it remains somewhat esoteric. The first aim of this paper is to investigate the preservation of Condition (A) under basic semigroup-theoretic constructions. Recently, Khosravi, Ershad and Sedaghatjoo have shown that every left S-act has a strongly flat or Condition (P) cover if and only if every cyclic left S-act has such a cover and Condition (A) holds. Here we find a range of classes of S-acts $\mathcal{C}$ such that every left S-act has a cover from $\mathcal{C}$ if and only if every cyclic left S-act does and Condition (A) holds. In doing so we find a further characterisation of Condition (A) purely in terms of the existence of covers of a certain kind. Finally, we make some observations concerning left perfect monoids and investigate a class of monoids close to being left perfect, which we name left$\mathcal{IP}$a-perfect.

scholarly journals THE ANNIHILATING-IDEAL GRAPH OF COMMUTATIVE RINGS I

2011 ◽  
Vol 10 (04) ◽  
pp. 727-739 ◽  
Author(s):  
M. BEHBOODI ◽  
Z. RAKEEI
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Connected Graph ◽  
Undirected Graph ◽  
Chain Condition ◽  
Commutative Rings ◽  
Finiteness Conditions ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

Let R be a commutative ring, with 𝔸(R) its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the annihilating-ideal graph of R, denoted by 𝔸𝔾(R). It is the (undirected) graph with vertices 𝔸(R)* ≔ 𝔸(R)\{(0)}, and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some finiteness conditions of 𝔸𝔾(R). For instance, it is shown that if R is not a domain, then 𝔸𝔾(R) has ascending chain condition (respectively, descending chain condition) on vertices if and only if R is Noetherian (respectively, Artinian). Moreover, the set of vertices of 𝔸𝔾(R) and the set of nonzero proper ideals of R have the same cardinality when R is either an Artinian or a decomposable ring. This yields for a ring R, 𝔸𝔾(R) has n vertices (n ≥ 1) if and only if R has only n nonzero proper ideals. Next, we study the connectivity of 𝔸𝔾(R). It is shown that 𝔸𝔾(R) is a connected graph and diam (𝔸𝔾)(R) ≤ 3 and if 𝔸𝔾(R) contains a cycle, then gr (𝔸𝔾(R)) ≤ 4. Also, rings R for which the graph 𝔸𝔾(R) is complete or star, are characterized, as well as rings R for which every vertex of 𝔸𝔾(R) is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

Sign in / Sign up

Export Citation Format

Share Document