Skew Generalized Power Series Rings

• A skew generalized power series ring is a noncommutative structure of functions from a strictly ordered monoid to a ring, with multiplication defined by a twisted convolution product.
• It generalizes constructions like skew polynomial, power series, Laurent series, and Malcev-Neumann series, providing a unified framework for algebraic and module-theoretic investigations.
• Its structural properties, including the left APP condition, are characterized using descending chain conditions on annihilators and right s-unital elements, impacting factorization and valuation theory.

A skew generalized power series ring is a noncommutative ring constructed as a space of functions from a strictly ordered monoid to a coefficient ring, endowed with pointwise addition and a convolution product twisted by a monoid homomorphism into the automorphism group of the ring. This construction intrinsically generalizes skew polynomial rings, skew power series rings, skew group rings, and Malcev-Neumann series, providing a unifying platform for both algebraic and homological investigations in the context of noncommutative ring theory, factorization, module theory, and valuation-theoretic properties.

1. Formal Definition and Construction

Let $R$ be a ring with unity, $(S, \leq)$ a strictly totally ordered monoid, and $\omega : S \to \mathrm{Aut}(R)$ a monoid homomorphism (notably, $\omega_{s+t} = \omega_s \omega_t$ and $\omega_1 = \mathrm{id}_R$). The skew generalized power series ring, denoted $[[R^{S, \leq}, \omega]]$ or $R[[S, \omega, \leq]]$, consists of all functions $f : S \to R$ with support

$\mathrm{supp}(f) = \{ s \in S \mid f(s) \ne 0 \}$

artinian (every strictly descending sequence is finite) and narrow (every set of pairwise non-comparable elements is finite).

Addition is defined pointwise, and the twisted convolution multiplication is given by

$$(fg)(t) = \sum_{(u, v) \in S^2,\, u + v = t} f(u)\,\omega_u(g(v))$$

where the sum is finite by the support conditions.

Embeddings:

• For $r \in R$, define $c_r(s) = r$ if $s = 1$, $0$ else.
• For $s \in S$, define $e_s(t) = 1$ if $t = s$, $0$ else.

These satisfy the relation $e_s c_r = c_{\omega_s(r)} e_s$.

This construction encompasses the following as special cases:

• Skew polynomial rings ($S = \mathbb{N}$, finite support)
• Skew power series rings ($S = \mathbb{N}$, infinite support)
• Skew Laurent polynomial/series rings ($S = \mathbb{Z}$)
• Skew group/monoid rings, Malcev-Neumann series (ordered groups/monoids)

2. Annihilator Properties and the Left APP Condition

A central structural focus is the behavior of annihilators in both the coefficient ring and its extension. For a left ideal $I \subseteq R$, right s-unitality is the property that for each $x \in I$ there exists $e \in I$ with $x = x e$. A ring $R$ is a left APP-ring if for any $a \in R$, the left annihilator

$l_R(Ra) = \{ r \in R \mid r a = 0 \}$

is right s-unital.

The paper establishes the following equivalence:

• If $(S, \leq)$ is a strictly totally ordered monoid, $w: S \to \mathrm{Aut}(R)$, and $R$ satisfies the descending chain condition (DCC) on right annihilators,
• then the skew generalized power series ring $[[R^{S, \leq}, w]]$ is left APP if and only if for every $S$-indexed subset $A \subseteq R$,

$$l_R\Bigg(\sum_{a \in A} \sum_{s \in S} R\,\omega_s(a)\Bigg)$$

is right s-unital.

The result deepens the connection between annihilator structure in $R$ and the behavior of ideals and modules in the skew extension, generalizing classical annihilator-based ring properties (left p.q.-Baer, PP, and PF-rings).

3. Main Results and Structural Theorems

Key results include:

Theorem 3 (Characterization of Left APP in Skew Generalized Power Series Rings)

Let $(S, \leq)$ be a strictly totally ordered monoid, $w: S \to \mathrm{Aut}(R)$ a monoid homomorphism, and $R$ satisfy DCC on right annihilators. The following are equivalent:

• $[[R^{S, \leq}, w]]$ is a left APP-ring;
• For any $S$-indexed subset $A$ of $R$, the ideal $$l_R\big(\sum_{a \in A}\sum_{s \in S} R \omega_s(a)\big)$$ is right s-unital.

The proof exploits minimality arguments with respect to DCC and transfinite induction, employing support properties of functions in the series ring and the twist structure of the convolution.

Notable corollaries:

• Formal power series rings $R[[x; \alpha]]$ and skew Laurent polynomial rings $R[x, x^{-1}; \alpha]$ are left APP under appropriate conditions.
• For the ring of arithmetical functions with Dirichlet convolution (when $S=(\mathbb{N}, \cdot)$), the left APP property is governed by the right s-unitality of annihilators in $R$ indexed by countable sets.

4. Applications, Examples, and Broader Context

Examples

• For $S = ( \mathbb{N}, \cdot )$ ($\leq$ the usual order), $[[R^{(\mathbb{N},\cdot)}, \leq]]$ models the ring of arithmetical functions with Dirichlet convolution.
• For $S = \mathbb{Z}$ and $w$ defined by two commuting automorphisms $\alpha, \beta$, the construction yields skew power series rings in multiple variables (e.g., $R[[x, y; \alpha, \beta]]$).

Applications

• Understanding when $[[R^{S, \leq}, w]]$ is left APP clarifies the transfer of annihilator properties and ideal-theoretic behavior from $R$ to its noncommutative extensions.
• This property plays a role in module decompositions, the analysis of stably free modules, and reveals purity of annihilators in extended settings.
• The results generalize the left APP property beyond skew polynomial and power series rings, enabling a systematic paper of more general noncommutative ring extensions.

Module and Ring-Theoretic Implications

The left APP property implies the preservation of certain module-theoretic and homological behaviors, such as pure submodules and ideal purity in module categories, and underlies structural results analogous to those for classical Baer and PF-rings.

5. Comparison with Related Ring Constructions

The skew generalized power series ring unifies and extends diverse constructions:

• Skew polynomial rings: by restricting to $S = \mathbb{N}$ and finite support.
• Malcev-Neumann series: by $S$ a totally ordered group.
• Classical commutative structures: by setting $w$ to the identity.

This abstraction allows common handling of properties (e.g., left APP, Baer, quasi-Baer) using unified support and twist conditions, generalizing known preservation results and illuminating new classes of noncommutative rings.

6. Generalizations and Theoretical Perspectives

Generalizations explored in the literature include:

• Relaxation of strictness on the monoid order (to quasi-totally ordered or artinian monoids), leading to different classes of series rings with more general support sets.
• Weaker hypotheses on $R$: Studying the impact of weaker chain conditions (e.g., between DCC on annihilators and ACC on principal ideals) on the APP and related properties in the extension.
• Connections to Baer, quasi-Baer, and S-Noetherian properties, revealing the interplay between support conditions, left/right annihilator properties, and module-theoretic behavior in the generalized and noncommutative framework.

Such generalizations provide a toolkit for advancing the structure theory of noncommutative rings, especially those arising as rings of functions, operator algebras, and power series rings with symmetries and automorphisms.

7. Impact and Synthesis

The criterion characterizing when a skew generalized power series ring is left APP provides a practical tool for determining the ideal structure and module-theoretic features of rings built through generalized power series and noncommutative twists. This organizes the understanding of extensions ranging from group rings to formal series, encompassing both previously known and novel classes of rings. The unifying approach to annihilator-related properties deepens the interplay between ring-theoretic, module-theoretic, and homological aspects in noncommutative algebra and supports further structural and computational investigations.