On the structural behavior of images of polynomials

Technical Summary: On the Structural Behavior of Images of Polynomials

Problem Statement
This paper investigates the structural properties of the images of noncommutative polynomials evaluated on associative algebras. Specifically, it focuses on the additive and multiplicative substructures generated by these images, including additive subgroups, subrings, and the representation of algebra elements as sums or products of polynomial values. The study is motivated by the L'vov-Kaplansky conjecture, which posits that the image of a multilinear polynomial on a matrix algebra over an infinite field is a vector space, and by recent inquiries into whether matrices can be expressed as products of additive commutators or differences of multiplicative commutators.

Methodology
The authors employ a combination of ring-theoretic techniques, including the theory of prime rings, extended centroids, and Martindale symmetric rings of quotients. The methodology involves:

• Ideal Theory: Analyzing the containment of nonzero ideals within sums of products of polynomial values on prime and simple rings.
• Commutator Analysis: Utilizing properties of Lie ideals and additive commutators to determine generation properties of rings.
• Matrix Decomposition: Applying block diagonal sums, similarity transformations, and the structure of triangular matrices (specifically unitriangular and nilpotent matrices) to decompose elements in matrix algebras over division rings.
• Factorization: Introducing "decomposable" multilinear polynomials (products of polynomials in disjoint variable sets) to simplify the analysis of their images.
• Geometric and Algebraic Arguments: In the context of real quaternions, the paper utilizes norm inequalities, inner products, and the specific geometry of the unit sphere to characterize differences of multiplicative commutators.

Key Contributions and Results

1. Additive Structures Generated by Polynomial Images:

   • Theorem 1: In a prime ring $R$ with extended centroid $C$, if a polynomial $f \in C\langle X \rangle$ is not central-valued on $RC$, then the additive subgroup generated by finite sums of products of two values of $f$ on a nonzero ideal contains a nonzero ideal of $R$. The only exception is when $R \cong M_2(\text{GF}(2))$.
   • Corollary 4: For a simple $F$-algebra $R$, if $p$ is not central-valued, the subring generated by the image $p(R)$ coincides with $R$ (up to the $M_2(\text{GF}(2))$ exception).
   • Theorem 6: If $f(R)$ is "fully noncentral" in an algebra over a field with more than two elements, then for every maximal ideal $M$, $R = (f(R)^2)_+ + M$.
   • Theorem 8 & 10: For unital algebras generated by commutators as ideals, if $R = (f(R)^n)_+$, then every element in $R$ can be written as a sum of a bounded number of elements from $f(R)^n$. This generalizes results on the generation of rings by commutators.

2. Products of Additive Commutators:

   • Theorem 17: If $D$ is a division ring with a finite center $F$ ($|F| = q > n$), every matrix in $M_n(D)$ is a product of two additive commutators.
   • Theorem 22: For any field $F$, there exists a noncommutative division ring $D$ with center $F$ (infinite-dimensional over $F$) where every nonzero element is a product of two noncentral commutators.
   • Theorem 23: Under specific conditions on the field size and matrix parity, if every nonzero element of a division ring $D$ is a product of two noncentral commutators, then every matrix in $M_n(D)$ is a product of two commutators.
   • Theorem 30: If $D$ is an algebraic division ring, every matrix in $M_n(D)$ is a product of two commutators.
   • Theorem 31: Every element in the finitary matrix algebra $M_\infty(D)$ is a product of two commutators.

3. Images of Decomposable Polynomials:

   • The paper introduces decomposable multilinear polynomials (products of polynomials in disjoint variables).
   • Theorem 34 & 35: For finitary matrix algebras over infinite fields (and division rings with infinite centers), the image of a decomposable multilinear polynomial $p = p_1 p_2$ (where $p_1, p_2$ are non-zero on the algebra) is the entire algebra.
   • Theorem 39: For $2 \times 2$ matrices over a division ring, if $p = p_1 p_2$ satisfies certain non-degeneracy conditions, the image is the whole algebra.

4. Polynomial Commutators:

   • Theorem 43 & Corollary 47: For a noncommutative infinite simple algebra $R$, the additive subgroup generated by polynomial commutators $p[R, R]$ (where $p(x) - p(yx)$ is not central-valued) equals the additive commutator subgroup $[R, R]$, which in turn equals $R$ (by Herstein's theorem). Thus, $R$ is generated as a ring by its polynomial commutators.

5. Multiplicative Commutators and Real Quaternions:

   • Theorem 49: In the real quaternion division ring $\mathbb{H}$, an element $q$ can be written as the difference of two elements of norm one (elements of $SL_1(\mathbb{H})$) if and only if $\|q\| \le 2$. This contrasts with the $n \ge 2$ case where $M_n(D) = SL_n(D) - SL_n(D)$.
   • Theorem 50 & 51: For $n=1$, the set of differences of multiplicative commutators from polynomial images is restricted to elements with norm $\le 2$. However, for $n \ge 2$, every matrix in $M_n(\mathbb{H})$ can be expressed as the difference of two products of multiplicative commutators derived from multilinear polynomial images.

Significance
The paper claims to provide a systematic investigation into the algebraic substructures generated by noncommutative polynomial images. It establishes that, with minor exceptions (specifically $M_2(\text{GF}(2))$), the images of noncentral polynomials on simple algebras are sufficiently rich to generate the entire algebra additively and multiplicatively. The work extends the understanding of the L'vov-Kaplansky conjecture to decomposable polynomials and resolves the question of expressing matrices as products of commutators in various settings, including division rings with finite centers and algebraic division rings. Furthermore, it highlights a distinct structural dichotomy between the one-dimensional case (real quaternions) and higher-dimensional matrix algebras regarding the representation of elements as differences of multiplicative commutators.