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

## TL;DR

This paper investigates the conditions under which certain subspaces, subalgebras, and ideals generated by polynomial values in an algebra are finite-dimensional, and explores dual problems involving finite-codimensional subspaces, especially for multilinear polynomials.

## Contribution

It introduces new results relating finite-dimensionality of polynomial-generated subspaces to that of subalgebras and ideals, and examines duality with finite-codimensional marginal subspaces for multilinear polynomials.

## Key findings

- Finite-dimensionality of the verbal subspace does not necessarily imply the same for the generated subalgebra and ideal.
- The paper establishes conditions under which these finite-dimensionality properties are equivalent.
- It explores the dual problem involving finite-codimensional marginal subspaces for multilinear polynomials.

## Abstract

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