Power-Serieswise McCoy Modules

Abstract

Extending the notion of McCoy modules and power-serieswise Armendariz modules, we introduce the concept of power-serieswise McCoy modules. Various basic properties of these modules are discussed and many illustrative examples are provided. Moreover, power-serieswise McCoy modules are used to obtain results on some polynomial extensions and matrix extensions of a module.

1. Introduction

Throughout the paper, all rings considered are associative with identity and modules are unitary right modules. Let be the set of natural integers. The symbol stands for the matrix unit. For a set of a module , stands for the right annihilator of in the ring . We write , and for the polynomial ring, the power-serieswise ring, the Laurent polynomial ring, and Laurent power-serieswise polynomial ring over , respectively. For a module , we consider Each of these is an abelian group under an obvious addition operation. Moreover, becomes an module under the following product operation: for , Similarly, and are -module, -module, and -module, respectively.

Chhawchharia, Rege [1], and Nielsen [2] independently called a ring right McCoy (respectively, left McCoy) if whenever for , there exists (respectively, ) such that (respectively, ). A ring is McCoy if it is both left and right McCoy. The term “McCoy ring" was coined because McCoy [3] had shown that every commutative ring satisfies the above-mentioned condition. Armendariz rings (i.e., if and in satisfy , then for each and ) are well known example of McCoy ring. For more details of rings with McCoy conditions, one may refer to [4–6], etc. The McCoy properties of rings were extended to power-serieswise rings in [7]. A ring is called a right power-serieswise McCoy ring if whenever satisfy , then there exists a nonzero element such that . A left power-serieswise McCoy can be defined similarly. A left and right power-serieswise McCoy is called power-serieswise McCoy. Recall that a ring is reduced [8] if it has no nonzero nilpotent elements. In [7], it was claimed that all reduced rings were power-serieswise McCoy and so are McCoy.

Due to [9], a module is Armendariz if for all whenever and satisfy . As both generalization of McCoy rings and Armendariz modules, McCoy modules were introduced in [10]. A module is called McCoy if where and , then there exists such that In [11], Lee and Zhou extended the concept of Armendariz modules to power-serieswise modules and introduced the concept of power-serieswise Armendariz modules. A module is said to be power-serieswise Armendariz if with , then for all and .

Motivated by the above, we introduce the notion of power-serieswise McCoy modules as a straightforward extensions of McCoy modules and power-serieswise McCoy rings. We show that the class of power-serieswise Armendariz modules are properly contained in the class of power-serieswise McCoy modules. Basic properties of this class of modules are investigated, and examples are provided. In addition, we study the relationship between a module and its polynomial module (respectively, matrix module) by means of power-serieswise McCoy properties.

2. Power-Serieswise McCoy Modules

Definition 1. Let be a module over a ring and the corresponding polynomial module over . is called power-serieswise McCoy if whenever where and , there exists a nonzero element such that .

It is clear that power-serieswise McCoy modules are McCoy and is power-serieswise McCoy iff is a right power-serieswise McCoy ring. Every submodule of a power-serieswise McCoy module is power-serieswise McCoy. In particular, if is a right ideal of a power-serieswise McCoy ring , then is a power-serieswise McCoy module.

The following result is motivated by [10, Proposition 2.3].

Proposition 2. Let be any nonzero ideal of a ring , then is a power-serieswise McCoy module.

Proof. Give any . Take a nonzero element . Since is an ideal of , . That is,

The following example reveals that McCoy module need not be power-serieswise McCoy.

Example 3 (see [7, Example 2.6]). Let be a field, the free algebra of polynomials in noncommuting indeterminates over , and the ideal of generated by with . Let Identify with for simplicity. Consider the equality with For any nonzero , if , then and . The first equality implies , but , a contradiction. Thus, is not power-serieswise McCoy. However, the ring is McCoy since it is Armendariz by [12, Example 2.1].

Theorem 4. Power-serieswise Armendariz modules are power-serieswise McCoy.

Proof. Let be a power-serieswise Armendariz module and , where ; there exists such that . Hence, for each which deduces that . Therefore, is a power-serieswise McCoy module.

Let . A module is p.p.-module [13] if for any . A module is called abelian [13] if for any and any , any idempotent , . In view of [13, Corollary 2.9] and Theorem 4, we have the following result.

Corollary 5. . Then for all . Since , if is an abelian and p.p.-module, then is power-serieswise McCoy.

The converse of Theorem 4 does not hold generally.

Example 6. Let be a reduced ring and let Then is not an Armendariz ring by [8, Example 3], so is not power-serieswise Armendariz. Since is reduced, is a power-serieswise McCoy ring by [7, Corollary 2.9].

A module is called reduced [11] if for any and any , implies that . All reduced modules are Armendariz and hence McCoy.

Remark 7. In [13], a ring is called strong regular if for any , there exists such that . By [14, Theorem 2.16], all modules over strong regular rings are reduced, so is power-serieswise Armendariz by [11, Lemma 1.5], and hence power-serieswise McCoy.

A module is semicommutative [9] if whenever and satisfy , then for every A module is said to be finitely cogenerated [15] if for every set of submodules of with , there is a finite subset (i.e., and is finite) with .

Cui and Chen [10] proved that a semicommutative module over a reduced ring is McCoy; we have the following result.

Proposition 8. Let be a reduced ring and a finitely cogenerated module. Then a semicommutative module is power-serieswise McCoy.

Proof. Let with Assume that (If not, set with a minimal such that , and we have ) This implies the following system of equations: Since is semicommutative, we have from Eq.. Multiplying on the right by yields . Similarly, multiplying on the right by gives because . Continuing this procedure, one has for all . Since is reduced, implies for each . Thus and imply . Since is a finitely cogenerated module, for some finite . Take . Then . Hence is power-serieswise McCoy.

A module is uniform [15] if any two nonzero submodules have a nonzero intersection. We have a similar result as in [10].

Theorem 9. Let be a family of McCoy -modules for an index set . Then
(1) if is uniform, then a direct sum is power-serieswise McCoy;
(2) if is an infinite set and is uniform and finitely cogenerated, then a direct product is power-serieswise McCoy.

Proof. If with and , let . Since is power-serieswise McCoy and , there exists such that for each .
We now prove (1). Note that the set is finite. Put . Since is uniform, we have Take any Then for each , whence Thus, is power-serieswise McCoy.
To show (2), the assumption is uniform implies that for any finite subset As is finite cogenerated, it follows that . Similar to the proof of (1), we are done.

Let be a module. Put If is a commutative domain, then is a submodule of called the torsion submodule of (see [15]).

Let be a module over a ring where . Set . It is clear that is an abelian group isomorphism via the usual map. So can be viewed as . Similarly, . Then can be viewed as . So we can define

Theorem 10. Let be a module over a ring where . If is a power-serieswise McCoy -module, then is a power-serieswise McCoy -module. The converse holds if for all .

Proof. Suppose that is a power-serieswise McCoy -module for any . Let where and . Set , . Then . By the discussion above, we have So for all . Since , there exists such that The assumption that is a power-serieswise McCoy -module implies that there exists such that . Take Then and .
Conversely, fix any . Let with where . Write and Then Since is power-serieswise McCoy -module, there exists such that . That is, If , then and imply . However, yields . Hence, is a right power-serieswise McCoy -module. Since is arbitrary, we have done.

Proposition 11. Let be a commutative domain and an -module. The module is power-serieswise McCoy if and only if its torsion submodule is power-serieswise McCoy.

Proof. Let and satisfy . Then We may assume since Now multiplying by on the right yields ; thus annihilates both and . Similarly, multiplying on the right by yields Continuing this procedure, one has for all . So , that is . As is power-serieswise McCoy, there exists such that So is power-serieswise McCoy. The other implication is trivial.

Recall that a classical right quotient ring for a ring is a ring which contains as a subring in such a way that every regular element (that is, nonzero-divisor) of is invertible in and .

Proposition 12. Suppose that there exists the classical right quotient ring of a ring and is a -module. Then is power-serieswise McCoy if and only if is power-serieswise McCoy.

Proof. Suppose that is a power-serieswise McCoy module. Let and with . Since is the classical right quotient ring of , by [16, Proposition 2.1.16] we may assume that with and some regular element . Set , then and . So . Since is power-serieswise McCoy, there exists such that . Therefore, is power-serieswise McCoy.
Conversely, note that is the subring of , so is power-serieswise McCoy.

3. Polynomial Modules

In this section, we study the extensions of power-serieswise McCoy module , such as and And the properties of special matrix extensions of a power-serieswise McCoy module are also considered.

Let be a right -module. If is a multiplicatively closed subset of central regular elements of , set then has an -module structure. is called -torsion free if whenever and , we have (see [17]).

Proposition 13. Let be an -torsion free right -module. Then is power-serieswise McCoy if and only if is power-serieswise McCoy as an -module.

Proof. Suppose that is a power-serieswise McCoy module. Let and satisfy . Thus where . Since is power-serieswise McCoy, there exists such that , so . Hence is power-serieswise McCoy.
Conversely, let satisfy . Since is power-serieswise McCoy, there exists such that , thus . So is power-serieswise McCoy.

Corollary 14. A ring is power-serieswise McCoy if and only if so is .

Theorem 15. Let be a module. Then the following statements are equivalent:(1) is a power-serieswise McCoy -module.(2) is a power-serieswise McCoy -module.(3) is a McCoy -module.(4) is a power-serieswise McCoy -module.(5) is a McCoy -module.

Proof. Some of the proof is similar to that of [10, Theorem 3.2].
Suppose that is power-serieswise McCoy. Let and with , where . The degree of zero polynomial is taken to be . Let If we set , then , and the set of coefficients of s (respectively, s) equals the set of coefficients of (respectively, ). Since the equation deduced that , there exists such that . Thus . Hence, is power-serieswise McCoy.
Assume that is power-serieswise McCoy and , where . Set and . Then and Since is power-serieswise McCoy, there exists such that Let be a nonzero coefficient of since , then , that is Hence is power-serieswise McCoy.
Suppose that is power-serieswise McCoy. Let and with , where for and . Note that Write where . Then Similarly, Write where . Then . Obviously, . By the assumption, is a power-serieswise McCoy -module. Thus there exists such that . Set . Then and . Hence is McCoy. The other complication can be proved similarly.
Let . Then is a multiplicatively closed subset of central regular elements of . Since , , we have done by Proposition 13.
It is similar to the proof of .

Corollary 16. Let be a ring. Then the following statements are equivalent:(1) is a power-serieswise McCoy ring.(2) is a power-serieswise McCoy ring.(3) is a McCoy ring.(4) is a power-serieswise McCoy ring.(5) is a McCoy ring.Let be a module. For , denotes the following subring of the ring of all matrices over : and denotes the following additive group: Then is an -module under the usual matrix addition operation and scalar multiplication.

Proposition 17. The -module is power-serieswise McCoy if and only if the -module is power-serieswise McCoy.

Proof. Suppose that is power-serieswise McCoy. Let satisfy , where Under the usual matrix addition operation, we haveand Now we proceed with the following cases.

Case 1. Suppose that . Take . Thus .

Case 2. Suppose that and . Since is power-serieswise McCoy, implies that there exists such that . Take , then .

Case 3. Suppose that and . Since , there must be a minimal index such that and , where . Consider the (, )-entry of matrix ; we have It follows from is power-serieswise McCoy that there exists such that . Take , then

Therefore, is a power-serieswise McCoy -module for all cases.

Conversely, let satisfy where . Take , where is the unit matrix. It is easy to see that . Since is a power-serieswise McCoy -module, there exists , such that . By , there exists (if , ) such that . Hence is power-serieswise McCoy.

Corollary 18 (see [7, Corollary 2.9]). is a power-serieswise McCoy ring if and only if so is where .

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Conflicts of Interest

The authors declare that they have no conflicts of interest.