Some results on finiteness of radical algebras

1970 ◽  
Vol 11 (3) ◽  
pp. 291-296
Author(s):  
Ahmad Mirbagheri
Keyword(s):  
Chain Condition ◽  
Finitely Generated ◽  
Radical Algebra ◽  
Finite Dimensional ◽  
Ascending Chain Condition ◽  
Algebra Over A Field

R denotes always a radical algebra over a field φ. A left ring ideal of R which is also a subvector space over φ is called a left algebra ideal of R. R is said to be left algebra noetherian if it satisfies the ascending chain condition for left algebra ideals. If dim R < ∞, then (i) R is finitely generated (ii) R is left alehra noetherian (iii) R is algebraic. Since the radical of an algebraic algebra is nil ([4] P. 19), conditions (i), (ii), (iii) are also sufficient for R to be finite-dimensional.

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.

Noetherian Banach Jordan pairs

2001 ◽  
Vol 130 (1) ◽  
pp. 25-36 ◽  
Author(s):  
N. BOUDI ◽  
H. MARHNINE ◽  
C. ZARHOUTI ◽  
A. FERNANDEZ LOPEZ ◽  
E. GARCIA RUS
Keyword(s):  
Recent Work ◽  
Associative Algebra ◽  
Jordan Algebra ◽  
Jacobson Radical ◽  
Alternative Algebra ◽  
Chain Condition ◽  
Finite Capacity ◽  
Jordan Pair ◽  
Finite Dimensional ◽  
Ascending Chain Condition

An associative or alternative algebra A is Noetherian if it satisfies the ascending chain condition on left ideals. Sinclair and Tullo [21] showed that a complex Noetherian Banach associative algebra is finite dimensional. This result was extended by Benslimane and Boudi [5] to the alternative case.For a Jordan algebra J or a Jordan pair V, the suitable Noetherian condition is the ascending chain condition on inner ideals. In a recent work Benslimane and Boudi [6] proved that a complex Noetherian Banach Jordan algebra is finite dimensional.Here we show the following results:(i) the Jacobson radical of a Noetherian Banach Jordan pair is finite dimensional;(ii) nondegenerate Noetherian Banach Jordan pairs have finite capacity;(iii) complex Noetherian Banach Jordan pairs are finite dimensional.

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.

scholarly journals THE HOCHSCHILD COHOMOLOGY RING MODULO NILPOTENCE OF A MONOMIAL ALGEBRA

2006 ◽  
Vol 05 (02) ◽  
pp. 153-192 ◽  
Author(s):  
EDWARD L. GREEN ◽  
NICOLE SNASHALL ◽  
ØYVIND SOLBERG
Keyword(s):  
Hochschild Cohomology ◽  
Cohomology Ring ◽  
Finite Dimensional Algebra ◽  
Finitely Generated ◽  
Dimensional Algebra ◽  
Monomial Algebra ◽  
Hochschild Cohomology Ring ◽  
Finite Dimensional ◽  
Algebra Over A Field ◽  
The Ideal

For a finite dimensional monomial algebra Λ over a field K we show that the Hochschild cohomology ring of Λ modulo the ideal generated by homogeneous nilpotent elements is a commutative finitely generated K-algebra of Krull dimension at most one. This was conjectured to be true for any finite dimensional algebra over a field in [13].

scholarly journals Projectivity of Hopf algebras over subalgebras with semilocal central localizations

2008 ◽  
Vol 2 (1) ◽  
pp. 1-40 ◽  
Author(s):  
Serge Skryabin
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Module Algebra ◽  
Finitely Generated ◽  
Residually Finite ◽  
Hopf Subalgebra ◽  
Finite Dimensional ◽  
Algebra Over A Field

AbstractThe purpose of this paper is to extend the class of pairs A, H where H is a Hopf algebra over a field and A a right coideal subalgebra for which H is proved to be either projective or flat as an A-module. The projectivity is obtained under the assumption that H is residually finite dimensional, A has semilocal localizations with respect to a central subring, and there exists a Hopf subalgebra B of H such that the antipode of B is bijective and B is a finitely generated A-module. The flatness of H over A is shown to hold when H is a directed union of residually finite dimensional Hopf subalgebras, and there exists a Hopf subalgebra of H whose center contains A. More general projectivity and flatness results are established for (co)equivariant modules over an H-(co)module algebra under similar assumptions.

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.

scholarly journals A Difference-Differential Basis Theorem

1970 ◽  
Vol 22 (6) ◽  
pp. 1224-1237 ◽  
Author(s):  
Richard M. Cohn
Keyword(s):  
Commutative Ring ◽  
Chain Condition ◽  
Finitely Generated ◽  
Hilbert Basis ◽  
Radical Ideal ◽  
Differential Basis ◽  
Basis Theorem ◽  
Finite Set ◽  
Radical Ideals ◽  
Ascending Chain Condition

Our aim in this paper is to extend to difference-differential rings the beautiful theorem of Kolchin [5, Theorem 3] for the differential case. The necessity portion of Kolchin's result is not obtained.What might well be called the Ritt basis theorem states that if a commutative ring R with identity is finitely generated over a subring R0, then the ascending chain condition for radical ideals of R0 implies the ascending chain condition for radical ideals of R. (This is indeed a basis theorem. If we define a basis for a radical ideal A to be a finite set B such that then every radical ideal of a ring R has a basis if and only if the ascending chain condition for radical ideals holds in R.) It is the Ritt basis theorem rather than the Hilbert basis theorem which has appropriate generalizations in differential and difference algebra, where in fact it originated.

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.

A note on (-1, 1) ring satisfying a.c.c

2014 ◽  
Vol 3 (2) ◽  
pp. 34
Author(s):  
Jayalakshmi K.
Keyword(s):  
Banach Algebra ◽  
Quotient Ring ◽  
Chain Condition ◽  
Direct Sums ◽  
Regular Elements ◽  
Finite Dimensional ◽  
Complex Banach Algebra ◽  
The Right ◽  
Ascending Chain Condition

Suppose that a semiprime (-1, 1) ring \(R\) is associative, satisfies the ascending chain condition for the right annihilators of the form \(r(w)\), where $w$ belongs to the nucleus \(N(R)\) and \(R\) contains no infinite direct sums of nonzero right ideals. Then the right quotient ring of $R$ relative to the subset \(W = \lbrace w \in N(R) / w \) is regular in \(R\rbrace\) exist and it is semisimple and artinian. Also if \(A\) be a nonassociative complex Banach algebra which satisfies ascending chain condition on left ideals and assume that the center \(Z(A)\) of \(A\) consists of regular elements then \(Z(A)\cong \mathbb{C}\). As a result if \(A\) be a (-1, 1) noetherian complex Banach algebra then \(A\) is finite-dimensional.

scholarly journals Interlacing methods and large indecomposables

1970 ◽  
Vol 3 (3) ◽  
pp. 337-348 ◽  
Author(s):  
S. E. Dickson ◽  
G. M. Kelly
Keyword(s):  
Artinian Ring ◽  
Finite Dimensional Algebra ◽  
Finitely Generated ◽  
Dimensional Algebra ◽  
Infinite Dimensional ◽  
Direct Sum ◽  
Number Of Generators ◽  
Finite Dimensional ◽  
Factor Module ◽  
Algebra Over A Field

The method of interlacing of modules, like amalgamation of groups, is a way of getting new objects from old. Briefly, the interlacing module we consider is a certain factor module of a direct sum of copies (finite or infinite) of an original module M. The conditions given in a previous paper by the first author in order that the interlacing module (using finitely many copies) be indecomposable are here greatly weakened, and we further allow the number of copies of the original to be infinite. R. Colby has shown that if R is a left artinian ring, the existence of a bound on the number of generators required for any indecomposable finitely-generated left R-module implies that R has a distributive lattice of two-sided ideals. This result is extended to rings whose identity is a sum of orthogonal local idempotents.For these rings the same distributivity is proved in case every indecomposable interlacing module of the above type which begins with an indecomposable projective M is finitely-generated. A consequence is that any finite-dimensional algebra over a field having infinitely many two-sided ideals has infinite-dimensional indecomposables.

Sign in / Sign up

Export Citation Format

Share Document