A ring is defined as a nonempty set equipped with two binary operations satisfying specific axioms [1]. Ribenboim [2] introduced the Generalized Power Series Rings (GPSRs) , extending the semigroup ring R[S] [3], the polynomial ring R[X], and the power series ring [4]. The GPSR imposes additional restrictions, requiring the support supp(f) to be both Artinian and narrow. A partially ordered set is called Artinian if every strictly ordered sequence of elements in S is finite, and narrow if every trivially ordered subset of S is finite [5].

Properties of GPSRs have been extensively studied [6, 7, 8, 9, 10, 11]. Varadarajan [12] introduced the Generalized Power Series Module (GPSM) , which generalizes the polynomial module over polynomial rings [13]. Further studies on GPSMs and their Noetherian properties can be found in [14, 15, 16].

Mazurek and Ziembowski [17] extended GPSRs by introducing a monoid homomorphism , forming the Skew Generalized Power Series Ring (SGPSR) . Subsequent studies [18, 19, 20, 21, 22] and related works [23, 24, 25, 26, 27, 28] investigated various algebraic properties of SGPSRs. Faisol et al. [29, 30] later constructed the Skew Generalized Power Series Module (SGPSM) , extending the theory of generalized power series to module structures over SGPSRs.

While much work has been devoted to GPSRs and SGPSRs, existing studies have predominantly focused on module structures over a single generalized or skew generalized power series ring. In many algebraic settings, however, it is natural to consider module structures involving two distinct skew generalized power series rings equipped with possibly diferent skewing homomorphisms. In such situations, the classical SGPSR-module framework is no longer suficient, since module actions and homomorphisms must simultaneously respect two independent skew structures.

Khumprapussorn et al. [31] generalized -bimodule structures by weakening compatibility conditions and introducing while homomorphisms of such modules were further studied by Yuwaningsih et al. [32]. Nevertheless, these works do not address the case where the underlying rings themselves are skew generalized power series rings. In particular, when considering module homomorphisms between modules over two distinct SGPSRs and , additional compatibility conditions arise from the simultaneous interaction of the two skewing homomorphisms ω and . These constraints do not appear in the classical single-ring SGPSR setting and cannot, in general, be reduced to previously studied cases.

This paper aims to fill this gap by developing a systematic framework for modules over two distinct skew generalized power series rings. In contrast to existing approaches confined to a single skew generalized power series ring, our construction captures bimodule-like structures governed by two independent skew actions. We investigate the resulting module homomorphisms and establish fundamental properties, including suficient conditions for elements to belong to their kernels, thereby extending skew generalized power series module theory beyond the classical single-ring setting.

To illustrate that the obtained results are not merely formal extensions, we also provide explicit and nontrivial examples demonstrating how module homomorphisms over two skew generalized power series rings induce concrete, coeficientwise kernel characterizations.

In this section we briefly recall the construction of skew generalized power series rings and their associated modules. These structures will serve as the algebraic framework used in the next section to construct a module.

Throughout this paper, denotes a strictly ordered monoid. Let be a ring with identity and let End be a monoid homomorphism. For a function , we write

Generalized power series rings were introduced by Ribenboim [2]. Given the homomorphism , the skew version developed by Mazurek and Ziembowski [17] is defined as

with multiplication

where the sum is finite. Here the twisting acts on the second factor of the convolution, and we refer to this product as the standard right skew convolution. If for all , one recovers the ordinary generalized power series ring.

For and , define the elements and in by

and

for every . These elements play the role of coeficient and basis elements in the skew generalized power series ring.

The corresponding module construction, considered for instance in [12, 29], is obtained as follows. Let M be a right -module and define

With pointwise addition, becomes a right -module under

where again the twisting occurs on the right factor of the convolution.

For and , define by

For a fixed , the assignment

defines an injective R^′-module homomorphism.

By analogy with the above construction, one may also consider a left module structure. Let R be another ring with identity and let End(R) be a monoid homomorphism. If M is a left R-module, then becomes a left ]-module under for and . Here the twisting acts on the left coeficient of the convolution, which is compatible with the left R-module structure of M.

These constructions provide the algebraic setting for combining two skew generalized power series rings, and , in order to construct the -module structure developed in the next section.

To proceed, we recall the definition of an -module as introduced in [31].

Definition 3.1.[31]Let R and be rings, and let M be an abelian group under addition. The structure M is called an -module if there exists a function

satisfying the following axioms:

(iii) r · m • (r^′ + t^′) = (r · m • r^′) + (r · m • t^′),

(iv)

for all , and

The following examples illustrate Definition 3.1.

Example 3.2 (Canonical example). Let R and be rings, and let M be an -bimodule in the classical sense. Define for all , and . Then M is an -module in the sense of Definition 3.1.

Example 3.3 (Non-bimodule example). Let and let . Define the ternary operation for all and . Then M is an -module that does not arise from a classical bimodule structure.

The first example shows that the notion of an -module extends the classical bimodule framework, while the second example demonstrates that the definition admits genuinely non-bimodule structures.

The following proposition records basic consequences of Definition 3.1, extending the axioms to finite sums in each variable.

Proposition 3.4. Let M be an -module. Then, for any , and , the following properties hold:

(iii.)

Proof. We prove each property as follows:

(i) For any , and

(ii) For any , and

(iii) Similar reasoning applies to , establishing the desired equality.

(iv) The distributive properties of the operation ensure the validity of the fourth property.

Throughout this paper, let R and be rings, a strictly ordered monoid, and let and be monoid homomorphisms.

For the left-hand ring , we adopt a right-twisted skew convolution compatible with the left module construction described in Section 2. Accordingly, multiplication in is defined by

for all

For the right-hand ring , we use the standard skew convolution recalled in Section 2. Thus multiplication is defined by

for all

If M is an -module as in Definition 3.1, we define

which is an abelian group under pointwise addition

for all

We now extend the -module structure of M to the level of skew generalized power series. To this end, we construct an operation combining the skew multiplications in and with the )-action on M given in Definition 3.1.

The construction is designed so that the twisting maps ω and act compatibly with the two-sided module structure, thereby producing a -module.

For any , and , define

We emphasize that the ternary operation introduced below is a primary construction. It is not obtained by iterating (5) and (6), but is designed to encode simultaneously the skew multiplications and the (R, R^′)-module action.

With this notation in place, define the ternary operation

by

Here the middle index y plays the role of a pivot. Both twisting maps ω and are evaluated at the same element y, ensuring compatibility of the two-sided action.

The following lemma ensures that the above operation is well defined.

Lemma 3.5. If , and , then

Proof. By assumption, the supports of , and are artinian and narrow subsets of . For any , the sum defining involves only triples such that supp(f), y ∈ supp(α), and with . Hence,

Since finite sums of artinian and narrow subsets of a strictly ordered monoid remain artinian and narrow (see [7]), it follows that supp is artinian and narrow. Therefore, f ∗ α ∗^′ □

The next theorem shows that the above construction yields a genuine module structure over two skew generalized power series rings.

Theorem 3.6.IfM is an , then the abelian group ,endowed with the operations defined in(7)and (9), is a -module.

Proof. To prove that is an -module, we verify the axioms in Definition 3.1. Let , and ;

(i) Additivity of α:

Therefore,

(ii) Additivity of f:

Hence,

(iii) Additivity of

Thus, ∗ α ∗^′ g^′.

(iv) Associativity: For any , we compute

By the definition of the ternary operation, each term is given by a finite sum over triples with . Substituting this expression and rearranging the finite sums, we obtain a sum over all tuples such that

Using the monoid homomorphism properties of and , namely and , together with the associativity of the product in , the summands can be regrouped to yield

where and with tb

Therefore,

which shows that the associativity axiom holds.

Example 3.7. Let be the additive monoid of nonnegative integers endowed with the natural order. Let and . Define homomorphisms

and

for all and

Then and are skew generalized power series rings with diferent twisting behaviours: the left ring has trivial twisting, while the right ring carries a nontrivial endomorphism action determined by

Let , viewed as an -module via the usual left and right scalar multiplications. Theorem the abelian group becomes a

In this case, for , and the ternary operation becomes

Since , the right action depends essentially on the skew structure induced

The following example provides an explicit computation illustrating the interaction between the two skew structures in the ternary operation.

Example 3.8. Let , and . Define and

Let be an -module with the usual scalar multiplications. Recall that the elements and in and and in are defined in (1) and (2), while the elements in are defined in(4).

Consider the elements

Since , the only triple contributing to is (1, 2, 1). Hence

Because , and , we obtain

Since and it follows that

This computation illustrates how the skew action determined by appears in convolution process.

The twisting map is defined on the coeficient ring , hence no support condition is involved at this level. The artinian and narrow support assumption appears only in the construction of the skew generalized power series ring and ensures that the sums arising in the multiplication and in the induced module operations are finite.

Example 3.9. Let and let be the additive monoid endowed with the product order. Define

Then is a monoid homomorphism, and the skew generalized power series ring is well defined.

This example shows that the theory applies to ordered monoids and twisting actions that difer substantially from those appearing in Example 3.7.

Examples 3.7 and 3.9 illustrate the robustness of the -module framework with respect to variations in both the twisting mechanisms and the underlying ordered monoid structures.

We begin by recalling the definition of an -module homomorphism, as introduced in [32].

Definition 4.1.Let M and N be -modules. An additive map is called an -module homomorphism if it satisfies

for all , and

Before defining homomorphisms between -modules, we establish the following lemma.

Lemma 4.2. Let M and N be -modules, and let and ]] be -modules. If is an homomorphism, then:

for all , and

Proof. (i) follows directly from the additivity of .

(ii) Let . By definition of the ternary operation, we have

where the sum is finite. Since is additive, we may apply termwise to this sum. Moreover, because is an -module homomorphism, it satisfies

Therefore,

Hence,

The following diagram describes how an -module homomorphism induces a homomorphism between the corresponding skew generalized power series modules:

Here and denote the canonical embeddings defined in (4), which identify each element with the skew generalized power series supported at a fixed Explicitly, With these identifications, the diagram is commutative; that is,

or equivalently,

Thus the assignment is compatible with module homomorphisms.

Using Lemma 4.2, we can extend the notion of an )-module homomorphism to the setting of -modules.

Theorem 4.3. Let M and N be , and let and be -modules. is an -module homomorphism, then there exists an -module homomorphism

Proof. Define a map for every as µ◦α. To prove σ is an )-module homomorphism, we verify the following:

(i) Additivity: For any ,

by Lemma 4.2(i)

(ii) Scalar Multiplication: For any , and ,

by Lemma 4.2(ii)

Thus, σ preserves addition and scalar multiplication. Therefore, σ is an -module homomorphism. □

Next, we define the kernel of an -module homomorphism, extending the notion of the kernel for -module homomorphisms introduced in [32].

Definition 4.4.Let be an module homomorphism. The kernel of σ is defined by

The following proposition gives a straightforward coeficientwise criterion for membership in the kernel of the induced homomorphism.

Proposition 4.5.Let be an -module homomorphism, and let be the induced -module homomorphism defined by

If satisfies then

Proof. If for all , then for every . Hence

Therefore

We conclude this section with examples illustrating the coeficientwise kernel criterion in diferent algebraic settings.

Example 4.6.Let and . Consider the Z-modules and define the -module homomorphism

Then

Let be a totally ordered monoid, for instance , and let be the homomorphism induced defined by

Define

Then for every , since . Hence, by Proposition 4.5 we obtain . Indeed, is the zero element of

This example shows that the kernel of the induced homomorphism is determined entirely coeficientwise. The presence of twisting maps does not afect this criterion, since the induced map σ acts pointwise on coeficients.

We now turn to situations where the twisting maps are nontrivial and the skew structure plays an essential role.

Example 4.7 (A genuinely skew example). Let and let k be a field. Set and . Define monoid homomorphisms for all . Then ω and are nontrivial homomorphisms from into End(R) and End , respectively.

For , the convolution product is given by

Thus each coeficient is shifted by i before multiplication, and the product depends essentially on the monoid index.

Therefore, the multiplication in cannot be reduced to the classical generalized power series multiplication. The same phenomenon occurs in , where coeficients are shifted in the y-variable. This provides a concrete commutative example in which the skew structure is genuinely present.

The previous example remains commutative at the level of coeficients. We next exhibit a genuinely noncommutative situation.

Example 4.8 (A noncommutative genuinely skew example). Let and let k be a field. Set , the ring of matrices over .

Fix an invertible matrix and define

for all n ∈ N and A, . Then ω and are monoid homomorphisms from into given by inner automorphisms.

For , the convolution product is

Thus each coeficient is conjugated by before multiplication with . Since matrix multiplication in is noncommutative, the order of twisting and multiplication is essential, and the product cannot be simplified to a classical form.

Similarly, in the coeficients are conjugated by , so that the and right twisting mechanisms act in opposite directions. Consequently, the two skew generalized power series rings interact in a genuinely noncommutative manner, reflecting the essential role ofinner automorphisms in the skew convolution structure.

Taken together, the preceding examples illustrate three distinct aspects of the theory. The first shows that the kernel of the induced homomorphism is determined coeficientwise. The second demonstrates that nontrivial twisting already produces skew behavior even when the coeficient rings are commutative. The third exhibits a genuinely noncommutative skew interaction arising from inner automorphisms. These examples clarify that the framework of two skew generalized power series rings extends the classical theory both at the level of module homomorphisms and at the level of convolution multiplication.

In this paper, we developed a systematic framework for modules over two distinct skew generalized power series rings. Starting from the notion of modules, we constructed the induced module endowed with a trilinear action

and proved that it satisfies the axioms of an -module. This construction extends the classical skew generalized power series module theory to a two-ring setting, providing a natural algebraic framework that captures bimodule-type behavior governed by two independent skew actions.

We further investigated the behavior of homomorphisms in this context. Specifically, we showed that every -module homomorphism induces a homomorphism between the corresponding modules over skew generalized power series rings. In addition, we established a coeficientwise suficient condition for elements of to belong to in terms of membership in ke . Although this characterization follows directly from the induced coeficientwise action, it confirms the structural compatibility of the construction within the two-ring skew framework. The examples provided illustrate how the presence of two independent skew actions influences the surrounding module structure.

More precisely, the main contributions of this paper consist of: (i) the construction of -module structures from -modules; (ii) the extension of -module homomorphisms to homomorphisms between the associated skew generalized power series modules; and (iii) a coeficientwise sufficient condition describing elements in the kernel of such induced homomorphisms.

The framework developed here provides a concrete algebraic foundation for further investigations of modules over two skew generalized power series rings. In particular, it ofers a natural setting for studying isomorphism theorems, quotient constructions, and exact sequences in this context, as well as for analyzing the transfer of structural properties (for example, finiteness conditions and Noetheriantype behavior) from )-modules to their associated skew generalized power series modules. These directions constitute natural avenues for future research.

PG.02.00.PL/2024