scholarly journals Chain conditions on composite Hurwitz series rings

2017 ◽  
Vol 15 (1) ◽  
pp. 1161-1170 ◽  
Author(s):  
Jung Wook Lim ◽  
Dong Yeol Oh
Keyword(s):  
Chain Condition ◽  
Polynomial Rings ◽  
Chain Conditions ◽  
Hurwitz Polynomial ◽  
Principal Ideals ◽  
Ascending Chain Condition

Abstract In this paper, we study chain conditions on composite Hurwitz series rings and composite Hurwitz polynomial rings. More precisely, we characterize when composite Hurwitz series rings and composite Hurwitz polynomial rings are Noetherian, S-Noetherian or satisfy the ascending chain condition on principal ideals.

Atomic rings and the ascending chain condition for principal ideals

1974 ◽  
Vol 75 (3) ◽  
pp. 321-329 ◽  
Author(s):  
Anne Grams
Keyword(s):  
Polynomial Ring ◽  
Chain Condition ◽  
Noetherian Rings ◽  
Unique Factorization ◽  
Maximum Condition ◽  
Maximum Element ◽  
Principal Ideals ◽  
Set Containment ◽  
Unique Factorization Domains ◽  
Ascending Chain Condition

LetRbe a commutative ring. We say thatRsatisfies theascending chain condition for principal ideals, or thatRhasproperty(M), if each ascending sequence (a1) ⊆ (a2) ⊆ … of principal ideals ofRterminates. Property (M) is equivalent to themaximum condition on principal ideals; that is, under the partial order of set containment, each collection of principal ideals ofRhas a maximum element. Noetherian rings, of course, have property (M), but the converse is not true; for ifRhas property (M) and if {Xλ} is a set of indeterminates overR, then the polynomial ringR[{Xλ}] has property (M). Krull domains, and hence unique factorization domains, have property (M).

scholarly journals On the semiprimitivity of skew polynomial rings

1993 ◽  
Vol 36 (2) ◽  
pp. 169-178 ◽  
Author(s):  
A. Moussavi
Keyword(s):  
Noetherian Ring ◽  
Chain Condition ◽  
Polynomial Rings ◽  
Skew Polynomial Rings ◽  
Skew Polynomial ◽  
Ascending Chain Condition

Let R be a left Noetherian ring with the ascending chain condition on right annihilators, let α be a ring monomorphism of R and δ an α-derivation of R. We prove that, if R is semiprime or α-prime, then R[X;α, δ] is semiprimitive (and left Goldie), and that J(R[X;α]) equals N(R)[X;α].

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.

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.

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.

scholarly journals Almost Krull rings

1986 ◽  
Vol 41 (2) ◽  
pp. 275-285
Author(s):  
E. Jespers ◽  
P. Wauters
Keyword(s):  
Group Ring ◽  
Polynomial Identity ◽  
Chain Condition ◽  
Polynomial Rings ◽  
Semi Group ◽  
Cyclic Subgroups ◽  
Skew Polynomial Rings ◽  
Krull Domain ◽  
Skew Polynomial ◽  
Ascending Chain Condition

AbstractThe notion of an almost Krull domain is extended to rings satisfying a polynomial identity. Some general structural results are obtained. We also prove that skew polynomial rings R [ X, σ] remain almost Krull if R is an almost Krull ring. Finally, we study when semigroup ring R[S] are almost Krull rings, in the case when the group of quotients of S has the ascending chain condition on cyclic subgroups. An example is included to show that the general (semi-) group ring case is much more difficult to deal with.

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