On the Nilpotency of Nil Subrings

1970 ◽  
Vol 22 (6) ◽  
pp. 1211-1216 ◽  
Author(s):  
Joe W. Fisher
Keyword(s):  
Left Ideal ◽  
Noetherian Ring ◽  
Short Proof ◽  
Sufficient Conditions ◽  
Chain Condition ◽  
Intermediate Step ◽  
Famous Theorem ◽  
Important Theorem ◽  
Left And Right ◽  
Ascending Chain Condition

A famous theorem of Levitzki states that in a left Noetherian ring each nil left ideal is nilpotent. Lanski [5] has extended Levitzki's theorem by proving that in a left Goldie ring each nil subring is nilpotent. Another important theorem in this area which is due to Herstein and Small [3] states that if a ring satisfies the ascending chain condition on both left and right annihilators, then each nil subring is nilpotent. We give a short proof of a theorem (Theorem 1.6) which yields both Lanski's theorem and Herstein- Small's theorem. We make use of the ascending chain condition on principal left annihilators in order to obtain, at an intermediate step, a theorem (Theorem 1.1) which produces sufficient conditions for a nil subring to be left T-nilpotent.

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].

scholarly journals QF-3 rings and torsion theories

1989 ◽  
Vol 46 (2) ◽  
pp. 251-261 ◽  
Author(s):  
José L. Gómez Pardo ◽  
Nieves Rodríguez González
Keyword(s):  
Chain Condition ◽  
Torsion Theory ◽  
Additional Hypothesis ◽  
Left And Right ◽  
Torsion Theories ◽  
Torsion Radical ◽  
Ascending Chain Condition

AbstractIn this paper, the rings which have a torsion theory τ with associated torsion radical τ such that R/t(R) has a minimal τ-torsionfree cogenerator are studied. When τ is the trivial torsion theory these are precisely the left QF-3 rings. For τ = τL, the Lambek torsion theory, this class of rings is wider but, with an additional hypothesis on τL it is shown that if R has this property with respect to the Lambek torsion theory on both sides, then R is a (left and right) QF-3 ring. The results obtained are applied to get new characterizations of QF-3 rings with the ascending chain condition on left annihilators.

scholarly journals Commutative rings whose quotients are Goldie

1975 ◽  
Vol 16 (1) ◽  
pp. 32-33 ◽  
Author(s):  
Victor P. Camillo
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Chain Condition ◽  
Commutative Rings ◽  
The Other ◽  
Direct Sums ◽  
Other Hand ◽  
Commutative Domain ◽  
Ascending Chain Condition

All rings considered here have units. A (non-commutative) ring is right Goldieif it has no infinite direct sums of right ideals and has the ascending chain condition on annihilator right ideals. A right ideal A is an annihilator if it is of the form {a ∈ R/xa = 0 for all x ∈ X}, where X is some subset of R. Naturally, any noetherian ring is Goldie, but so is any commutative domain, so that the converse is not true. On the other hand, since any quotient ring of a noetherian ring is noetherian, it is true that every quotient is Goldie. A reasonable question therefore is the following: must a ring, such that every quotient ring is Goldie, be noetherian? We prove the following theorem:Theorem. A commutative ring is noetherian if and only if every quotient is Goldie.

On a class of Noetherian algebras

1999 ◽  
Vol 129 (6) ◽  
pp. 1185-1196 ◽  
Author(s):  
E. Jespers ◽  
J. Okniński
Keyword(s):  
Finite Group ◽  
Homomorphic Image ◽  
Chain Condition ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Finitely Generated ◽  
Nilpotent Semigroup ◽  
Left And Right ◽  
Ascending Chain Condition

A class of Noetherian semigroup algebrasK[S]is described. In particular, we show that, for any submonoidSof the semigroupMnof all monomialn × nmatrices over a polycyclic-by-finite groupG, K[S]is right Noetherian if and only ifSsatisfies the ascending chain condition on right ideals. This is then used to prove that every prime homomorphic image of a semigroup algebra of a finitely generated Malcev nilpotent semigroupSsatisfying the ascending chain condition on right ideals is left and right Noetherian.

A note on the ascending chain condition of ideals

2019 ◽  
Vol 19 (07) ◽  
pp. 2050135 ◽  
Author(s):  
Ibrahim Al-Ayyoub ◽  
Malik Jaradat ◽  
Khaldoun Al-Zoubi
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Chain Condition ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
Regular Ideal ◽  
Reduction Number ◽  
Number Formula ◽  
A Chain ◽  
Ascending Chain Condition

We construct ascending chains of ideals in a commutative Noetherian ring [Formula: see text] that reach arbitrary long sequences of equalities, however the chain does not become stationary at that point. For a regular ideal [Formula: see text] in [Formula: see text], the Ratliff–Rush reduction number [Formula: see text] of [Formula: see text] is the smallest positive integer [Formula: see text] at which the chain [Formula: see text] becomes stationary. We construct ideals [Formula: see text] so that such a chain reaches an arbitrary long sequence of equalities but [Formula: see text] is not being reached yet.

scholarly journals Semigroups whose right ideals are finitely generated

2021 ◽  
pp. 1-36
Author(s):  
Craig Miller
Keyword(s):  
Sufficient Conditions ◽  
Chain Condition ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Equivalent Formulation ◽  
Regular Semigroups ◽  
Strong Semilattice ◽  
Necessary And Sufficient ◽  
Ascending Chain Condition ◽  
Completely Simple

Abstract We call a semigroup $S$ weakly right noetherian if every right ideal of $S$ is finitely generated; equivalently, $S$ satisfies the ascending chain condition on right ideals. We provide an equivalent formulation of the property of being weakly right noetherian in terms of principal right ideals, and we also characterize weakly right noetherian monoids in terms of their acts. We investigate the behaviour of the property of being weakly right noetherian under quotients, subsemigroups and various semigroup-theoretic constructions. In particular, we find necessary and sufficient conditions for the direct product of two semigroups to be weakly right noetherian. We characterize weakly right noetherian regular semigroups in terms of their idempotents. We also find necessary and sufficient conditions for a strong semilattice of completely simple semigroups to be weakly right noetherian. Finally, we prove that a commutative semigroup $S$ with finitely many archimedean components is weakly (right) noetherian if and only if $S/\mathcal {H}$ is finitely generated.

Study of skew inverse Laurent series rings

2017 ◽  
Vol 16 (12) ◽  
pp. 1750221 ◽  
Author(s):  
Kamal Paykan ◽  
Ahmad Moussavi
Keyword(s):  
Laurent Series ◽  
Present Note ◽  
Sufficient Conditions ◽  
Chain Condition ◽  
Power Series Ring ◽  
Series Ring ◽  
Derivation Formula ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Ascending Chain Condition

In the present note, we continue the study of skew inverse Laurent series ring [Formula: see text] and skew inverse power series ring [Formula: see text], where [Formula: see text] is a ring equipped with an automorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. Necessary and sufficient conditions are obtained for [Formula: see text] to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, projective-free and [Formula: see text]-ring, respectively. It is shown here that [Formula: see text] (respectively [Formula: see text]) is a domain satisfying the ascending chain condition (Acc) on principal left (respectively right) ideals if and only if so does [Formula: see text]. Also, we investigate the problem when a skew inverse Laurent series ring [Formula: see text] has the same Goldie rank as the ring [Formula: see text] and is proved that, if [Formula: see text] is a semiprime right Goldie ring, then [Formula: see text] is semiprimitive. Furthermore, we study on the relationship between the simplicity, semiprimeness, quasi-Baerness and Baerness property of a ring [Formula: see text] and these of the skew inverse Laurent series ring. Finally, we consider the problem of determining when [Formula: see text] is nilpotent.

scholarly journals A note on Noetherian orders in Artinian rings

1979 ◽  
Vol 20 (2) ◽  
pp. 125-128 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Triangular Matrix ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Regular Elements ◽  
Zero Divisors ◽  
Left And Right

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

Completely Indecomposable Modules

1949 ◽  
Vol 1 (2) ◽  
pp. 125-152 ◽  
Author(s):  
Ernst Snapper
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Chain Condition ◽  
Indecomposable Module ◽  
Unit Element ◽  
Indecomposable Modules ◽  
Operator Domain ◽  
Unit Operator ◽  
Ascending Chain Condition

The purpose of this paper is to investigate completely indecomposable modules. A completely indecomposable module is an additive abelian group with a ring A as operator domain, where the following four conditions are satisfied.1-1. A is a commutative ring and has a unit element which is unit operator for .1-2. The submodules of satisfy the ascending chain condition. (Submodule will always mean invariant submodule.)

On Duality Relative to Self-orthogonal Bimodules

2005 ◽  
Vol 12 (02) ◽  
pp. 319-332
Author(s):  
Jun Wu ◽  
Wenting Tong
Keyword(s):  
Noetherian Ring ◽  
Injective Dimension ◽  
Left And Right

Let R be a left and right Noetherian ring and RωR a faithfully balanced self-orthogonal bimodule. We introduce the notions of special embeddings and modules of ω-D-class n, and then give some characterizations of them. As an application, we study the properties of RωR with finite injective dimension. Our results extend the main results in [4].

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