Relationships between the canonical ascending and descending central series of ideals of an associative algebra

Riley, David M.; Shahada, Mayada

doi:10.1080/00927872.2016.1226871

Cited by 1 publication

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“…However, the following construction taken from [6] shows that Problem C has a negative solution for general algebras A, even for 'nice' polynomials f . Recall that the lower central series of A, when viewed as a Lie algebra, is defined by γ 1 (A) = A and γ n+1 (A) = [γ n (A), A], for all n ≥ 1.…”

Section: Corresponding Verbal and Marginal Subspacesmentioning

confidence: 99%

“…Recall that the lower central series of A, when viewed as a Lie algebra, is defined by γ 1 (A) = A and γ n+1 (A) = [γ n (A), A], for all n ≥ 1. [6]) Let K(α) be any simple field extension of a base field K, and let V be a vector space over K(α) with basis {v 1 , . .…”

Section: Corresponding Verbal and Marginal Subspacesmentioning

confidence: 99%

“…Example 4.3. (see Section 7 in [6]) Let K(α) be any simple field extension of a base field K, and let V be a vector space over K(α) with basis {v 1 , . .…”

Section: Problem Bmentioning

confidence: 99%

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On polynomials that are not quite an identity on an associative algebra

2019

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Let f be a polynomial in the free algebra over a field K, and let A be a K-algebra. We denote by SA(f ), AA(f ) and IA(f ), respectively, the 'verbal' subspace, subalgebra, and ideal, in A, generated by the set of all f -values in A. We begin by studying the following problem: if SA(f ) is finite-dimensional, is it true that AA(f ) and IA(f ) are also finite-dimensional? We then consider the dual to this problem for 'marginal' subspaces that are finite-codimensional in A. If f is multilinear, the marginal subspace, SA(f ), of f in A is the set of all elements z in A such that f evaluates to 0 whenever any of the indeterminates in f is evaluated to z. We conclude by discussing the relationship between the finite-dimensionality of SA(f ) and the finite-codimensionality of SA(f ). Verbal subspaces, subalgebras and idealsThroughout this paper, the term 'algebra' will be reserved for a not necessarily unital associative algebra A over a fixed base field K of characteristic p ≥ 0. We shall use A 1 to indicate its unital hull.Definition 1.1. Let A be an algebra, and let f = f (x 1 , . . . , x n ) be a polynomial in the free algebra K X on the set X = {x 1 , x 2 , . . .}. We shall denote by S A (f ), A A (f ) and I A (f ), respectively, the subspace, subalgebra and ideal in A generated by the set of all f -values in A:{f (a 1 , . . . , a n ) : a 1 , . . . , a n ∈ A}.We shall call the subspace S A (f ) the verbal subspace of A generated by f , and so forth.

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Section: Corresponding Verbal and Marginal Subspacesmentioning

confidence: 99%

Section: Corresponding Verbal and Marginal Subspacesmentioning

confidence: 99%