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

2018/12/19 by Jespers, Eric, Riley, David, Shahada, Mayada
#16R99 and 16S34 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1812.08205

Abstract

Let f be a polynomial in the free algebra over a field K, and let A be a K-algebra. We denote by §_A(f), \A_A(f) and \I_A(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 §<sub>A</sub>(f) is finite-dimensional, is it true that \A<sub>A</sub>(f) and \I<sub>A</sub>(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, \widehat§_A(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 §_A(f) and the finite-codimensionality of \widehat§_A(f).

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