On the uniform dimension of subextensions in skew polynomial rings

Technical Summary: On the Uniform Dimension of Subextensions in Skew Polynomial Rings

Problem Statement
The paper investigates the invariance of the uniform (Goldie) dimension and related concepts, such as the preservation of Goldie conditions and (semi)primeness, for subextensions within skew polynomial rings. Specifically, it addresses the behavior of subrings $A$ generated over a ground ring $R$ by subsets of standard terms within an ambient skew polynomial ring $S = R[X; \alpha, \delta]$ or a skew Laurent polynomial ring $S = R[X^{\pm 1}; \alpha]$.

The central problem is to determine under what conditions on the shape of a subextension $A$ the equality $\text{udim}(A) = \text{udim}(S) = \text{udim}(R)$ holds, and whether $A$ inherits the left Goldie property (finite uniform dimension and ascending chain condition on left annihilators) from $R$. The author notes that while this invariance is well-established for classical polynomial rings and iterated Ore extensions, the situation for arbitrary "polynomial-like" subrings in skew settings is complex. In non-commuting contexts, subrings generated by standard terms can behave "oddly," potentially failing to preserve uniform dimension or Goldie conditions, as demonstrated by examples where a prime left Goldie skew polynomial ring contains a monoid subring isomorphic to a free non-commutative polynomial ring (which has infinite uniform dimension).

Methodology
The paper employs a structural approach combining module-theoretic characterizations with specific algebraic constructions tailored to skew polynomial rings.

1. General Ring Extension Strategy: The author first establishes a general framework for arbitrary ring extensions $R \subset S$. A key concept introduced is that of a nicely essential subring (Definition 2.5). A subring $A$ is nicely essential in $S$ if for every finite set of non-zero elements in $S$, there exists a non-zero element in $A$ that maps all elements of the set into $A$. The paper proves that if $A$ is a nicely essential subring of $S$, they share the same uniform dimension, and properties like (semi)primeness and the Goldie condition are preserved under specific conditions (Propositions 2.6 and 2.7).

2. Commuting Variables Case: For skew Laurent polynomial rings with commuting variables, the paper utilizes the fact that any subextension generated by standard terms is a skew monoid ring over a submonoid of a free abelian group. By embedding these into a larger skew Laurent ring and utilizing the properties of denominator sets and Ore localization, the author demonstrates that such subextensions are nicely essential extensions of the coefficient ring, thereby preserving uniform dimension and Goldie properties (Theorems 3.3 and 3.5).

3. Non-Commuting Variables and "Essentially Special" Subextensions: In the general non-commuting case, the paper introduces the class of essentially special subextensions (Definition 4.7). These are subrings that are "nicely essential" within a "special subextension." A special subextension is defined by a specific structural shape involving attainable degrees of variables and induced automorphisms on coefficient subrings.

   • The methodology relies heavily on the concept of $R$-nice polynomials (Definition 4.4), where coefficients share a common left annihilator.
   • Technical lemmas (4.5, 4.10, 4.11) are developed to show that under the "special" shape, one can lift uniform and essential left ideals from the ground ring $R$ to the subextension $A$.
   • The proofs utilize transfinite induction on the well-ordered set of variables and reverse lexicographic ordering to handle multivariate cases.

Key Contributions and Results

• General Invariance Strategy: The paper provides a general strategy (Proposition 2.4) for comparing uniform dimensions in arbitrary ring extensions, removing the tacit assumption of finite uniform dimension found in earlier works (e.g., Shock [Sho72]). It characterizes modules with "enough uniform submodules" via essential direct sums (Proposition 2.3).
• Commuting Variables: The paper completely solves the problem for subextensions of skew Laurent polynomial rings with commuting variables. It proves that for any subextension $A$ generated by a family of standard terms, $\text{udim}(A) = \text{udim}(R)$, and $A$ is left Goldie (resp. (semi)prime) if and only if $R$ is (Theorems 3.3 and 3.5).
• Non-Commuting Variables (Essentially Special Subextensions): The main result (Theorem 4.15) establishes that for essentially special subextensions $A$ of a bijective skew polynomial ring $S$:
  • If $R$ has enough uniform left ideals, then $A$ has enough uniform left ideals and $\text{udim}(A) = \text{udim}(R)$.
  • The left singular ideal satisfies $\mathfrak{z}(A) = A \cap \mathfrak{z}(R)T$, where $T$ is the set of standard terms.
  • If $R$ is a (semi)prime left Goldie ring and $A$ is a strongly essentially special subextension, then $A$ is also a (semi)prime left Goldie ring with the same uniform dimension (Theorem 4.17).
• Counter-Examples: The paper provides explicit examples (Examples 2.9 and 2.10) of (semi)prime left Goldie skew polynomial rings containing monoid subrings isomorphic to free non-commutative polynomial rings. These subrings have infinite uniform dimension, demonstrating that the invariance of uniform dimension does not hold for all subextensions, thus highlighting the necessity of the "essentially special" condition.

Significance and Scope
The paper claims to extend classical results on the invariance of uniform dimension (such as those by Shock [Sho72], Matczuk [Mat95], and others) to a broader, more complex setting of subextensions in skew polynomial rings. Its significance lies in:

1. Handling Infinite Dimensions: Unlike previous works that often assumed finite uniform dimension, this paper allows for non-necessarily finite uniform dimensions.
2. Generalizing Subextensions: It moves beyond standard iterated Ore extensions to consider arbitrary subextensions generated by subsets of standard terms, identifying a specific class ("essentially special") where classical invariance properties hold.
3. Sharpness: The results are presented as sharp; the paper demonstrates through examples that without the "essentially special" condition, the preservation of uniform dimension and Goldie properties fails, even in well-behaved ambient rings.

The work does not propose new applications or experimental proposals but rather provides a rigorous theoretical framework for understanding the structural stability of Goldie conditions in non-commutative polynomial-like rings.