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

Abstract

Let f be a polynomial in the free algebra over a field K, and let A be a K-algebra. We denote by $\mathcal{S}$_A(f), $\mathcal{A}$_A(f) and $\mathcal{I}$_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 $\mathcal{S}$_A(f) is finite-dimensional, is it true that $\mathcal{A}$_A(f) and $\mathcal{I}$_A(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, $\widehat {\mathcal{S}}$_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 $\mathcal{S}$_A(f) and the finite-codimensionality of $\widehat {\mathcal{S}}$_A(f).