Nil ideals of finite codimension in alternative Noetherian algebras

In 1950, R. C. James [7] exhibited a nonreflexive Banach space with a basis that is of finite codimension in its second dual. This space is the first example of a quasi-reflexive space. General results on quasi-reflexive spaces have been obtained by P. Civin and B. Yood [3], and quasi-reflexive spaces with bases have been studied by D. Dean, B. L. Lin, and I. Singer [4 ; 12].

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Purity And Copurity in Systems of Linear Transformations

Canadian Journal of Mathematics ◽

10.4153/cjm-1976-086-7 ◽

1976 ◽

Vol 28 (4) ◽

pp. 889-896

Author(s):

Frank Zorzitto

Keyword(s):

Vector Spaces ◽

Finite Codimension ◽

Linear Transformations ◽

Complex Vector ◽

Finite Dimensional ◽

Quotient System

Consider a system of N linear transformations A1, … , AN: V → W, where F and IF are complex vector spaces. Denote it for short by (F, W). A pair of subspaces X ⊂ V, Y ⊂ W such that determines a subsystem (X, Y) and a quotient system (V/X, W/Y) (with the induced transformations). The subsystem (X, Y) is of finite codimension in (V, W) if and only if V/X and W / Y are finite-dimensional. It is a direct summand of (V, W) in case there exist supplementary subspaces P of X in F and Q of F in IF such that (P, Q) is a subsystem.

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Simple Quotients of Euclidean Lie Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1970-095-3 ◽

1970 ◽

Vol 22 (4) ◽

pp. 839-846 ◽

Cited By ~ 6

Author(s):

Robert V. Moody

Keyword(s):

Weyl Group ◽

Lie Algebras ◽

Bilinear Form ◽

Characteristic Zero ◽

Chain Condition ◽

Cartan Matrix ◽

Finite Codimension ◽

Infinite Dimensional ◽

Object Of Study ◽

Ascending Chain Condition

In [2], we considered a class of Lie algebras generalizing the classical simple Lie algebras. Using a field Φ of characteristic zero and a square matrix (Aij) of integers with the properties (1) Aii = 2, (2) Aij ≦ 0 if i ≠ j, (3) Aij = 0 if and only if Ajt = 0, and (4) is symmetric for some appropriate non-zero rational a Lie algebra E = E((Aij)) over Φ can be constructed, together with the usual accoutrements: a root system, invariant bilinear form, and Weyl group.For indecomposable (A ij), E is simple except when (Aij) is singular and removal of any row and corresponding column of (Aij) leaves a Cartan matrix. The non-simple Es, Euclidean Lie algebras, were our object of study in [3] as well as in the present paper. They are infinite-dimensional, have ascending chain condition on ideals, and proper ideals are of finite codimension.

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Vector bundles of finite rank on complete intersections of finite codimension in ind-Grassmannians

Complex Manifolds ◽

10.1515/coma-2015-0007 ◽

2015 ◽

Vol 2 (1) ◽

Author(s):

Svetlana Ermakova

Keyword(s):

Vector Bundle ◽

Complete Intersection ◽

Finite Rank ◽

Vector Bundles ◽

Complete Intersections ◽

Finite Codimension ◽

Line Bundles ◽

Direct Sum ◽

Rank Vector

AbstractIn this article we establish an analogue of the Barth-Van de Ven-Tyurin-Sato theorem.We prove that a finite rank vector bundle on a complete intersection of finite codimension in a linear ind-Grassmannian is isomorphic to a direct sum of line bundles.

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Residually finite dimensional algebras and polynomial almost identities

Journal of Algebra and Its Applications ◽

10.1142/s0219498822500384 ◽

2020 ◽

pp. 2250038

Author(s):

Michael Larsen ◽

Aner Shalev

Keyword(s):

Lie Algebra ◽

Finite Index ◽

Open Group ◽

Finite Codimension ◽

Finite Dimensional Algebra ◽

Nilpotent Ideal ◽

Dimensional Algebra ◽

Residually Finite ◽

Finite Dimensional ◽

Theoretic Problem

Let [Formula: see text] be a residually finite dimensional algebra (not necessarily associative) over a field [Formula: see text]. Suppose first that [Formula: see text] is algebraically closed. We show that if [Formula: see text] satisfies a homogeneous almost identity [Formula: see text], then [Formula: see text] has an ideal of finite codimension satisfying the identity [Formula: see text]. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra [Formula: see text] over [Formula: see text] is almost [Formula: see text]-Engel, then [Formula: see text] has a nilpotent (respectively, locally nilpotent) ideal of finite codimension if char [Formula: see text] (respectively, char [Formula: see text]). Next, suppose that [Formula: see text] is finite (so [Formula: see text] is residually finite). We prove that, if [Formula: see text] satisfies a homogeneous probabilistic identity [Formula: see text], then [Formula: see text] is a coset identity of [Formula: see text]. Moreover, if [Formula: see text] is multilinear, then [Formula: see text] is an identity of some finite index ideal of [Formula: see text]. Along the way we show that if [Formula: see text] has degree [Formula: see text], and [Formula: see text] is a finite [Formula: see text]-algebra such that the probability that [Formula: see text] (where [Formula: see text] are randomly chosen) is at least [Formula: see text], then [Formula: see text] is an identity of [Formula: see text]. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon,

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A Note on the Adjoint of the Product of Operators

Canadian Mathematical Bulletin ◽

10.4153/cmb-1970-007-6 ◽

1970 ◽

Vol 13 (1) ◽

pp. 39-45

Author(s):

Chia-Shiang Lin

Keyword(s):

Hilbert Space ◽

Linear Operator ◽

Closed Subspace ◽

Linear Operators ◽

Finite Codimension ◽

Restricted Condition ◽

Closed Linear Operators

Cordes and Labrousse ([2] p. 697), and Kaniel and Schechter ([6] p. 429) showed that if S and T are domain-dense closed linear operators on a Hilbert space H into itself, the range of S is closed in H and the codimension of the range of S is finite, then, (TS)* = S*T*. With a somewhat different approach and more restricted condition on S, the same assertion was obtained by Holland [5] recently, that S is a bounded everywhere-defined linear operator whose range is a closed subspace of finite codimension in H.

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