Algebra
Volume 2015 (2015), Article ID 595274, 4 pages
http://dx.doi.org/10.1155/2015/595274

Research Article

Generalized Malcev-Neumann Series Modules with the Beachy-Blair Condition

Mohamed A. Farahat1,2

¹Department of Mathematics and Statistics, Faculty of Science, Taif University, P.O. Box 888, Al-Hawiyah, Taif 21974, Saudi Arabia
²Mathematics Department, Faculty of Science, Al-Azhar University, P.O. Box 11884, Nasr City, Cairo, Egypt

Received 9 January 2015; Revised 9 March 2015; Accepted 10 March 2015

Academic Editor: Andrei V. Kelarev

Copyright © 2015 Mohamed A. Farahat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce a new class of extension rings called the generalized Malcev-Neumann series ring  with coefficients in a ring and exponents in a strictly ordered monoid which extends the usual construction of Malcev-Neumann series rings. Ouyang et al. in 2014 introduced the modules with the Beachy-Blair condition as follows: A right -module satisfies the right Beachy-Blair condition if each of its faithful submodules is cofaithful. In this paper, we study the relationship between the right Beachy-Blair condition of a right -module and its Malcev-Neumann series module extension .

1. Introduction

Throughout this paper denotes an associative ring with identity; is a strictly ordered monoid (i.e., is an ordered monoid satisfying the conditions that if , then and for . Recall that a subset of is said to be artinian if every strictly decreasing sequence of elements of is finite and that is narrow if every subset of pairwise order-incomparable elements of is finite. Suppose the two maps and (the group of invertible elements of ). Let denote the set of all formal sums such that supp is an artinian and narrow subset of , with componentwise addition and the multiplication rule is given by

for each and . In order to ensure the associativity, it is necessary to impose two additional conditions on and : namely, for all ,

(i),(ii), where denotes the automorphism of defined by

It is now routine to check that is a ring which is called the ring of generalized Malcev-Neumann series. We can assume that the identity element of is ; this means that

In this case is an embedding of as a subring into .

For each we denote by the set of minimal elements of supp. If is a strictly totally ordered monoid, then supp is a nonempty well-ordered subset of and consists of only one element.

Clearly, the above construction generalizes the construction of Malcev-Neumann series rings, in case of (an ordered group), which was introduced independently by Malcev and Neumann (see [1, 2]).

If the order is the trivial order, then is the usual crossed product ring . Also, if the monoid has the trivial order and is trivial, then is the usual skew monoid ring . However if the monoid has the trivial order and is trivial, then is the usual twisted monoid ring . Finally, if the monoid has the trivial order and and are trivial, then is the usual monoid ring (see Sections 3.2 and  3.3 in [3]).

Moreover, if is a ring endomorphism of , set endowed with the trivial order. Define via for every and for any . We have is the usual skew polynomial ring . However if is the usual order, then is the usual skew power series ring . If is a ring automorphism of and is the usual order, then is the usual ring of skew Laurent power series .

At the same time, if we set also for all , then it is easy to check that polynomial rings, Laurent polynomial rings, formal power series rings, and Laurent power series rings are special cases of .

If is a unitary right -module, then the Malcev-Neumann series module is the set of all formal sums with coefficients in and artinian and narrow supports, with pointwise addition and scalar multiplication rule is defined by

where and . One can easily check that and ensure that is a unitary right -module. For each we denote by the set of minimal elements of supp. If is a strictly totally ordered monoid, then supp is a nonempty well-ordered subset of and consists of only one element.

Recall from Faith [4] that a ring is called a right zip ring and if the right annihilator of a subset is zero, then for a finite subset of . Although the concept of zip rings was initiated by Zelmanowitz [5] it was not called so at that time.

Recall from [6] that a right -module is called a right zip module provided that if the right annihilator of a subset of is zero, then there exists a finite subset such that .

According to Rodríguez-Jorge [7], a ring satisfies the right Beachy-Blair condition if its faithful right ideals are cofaithful; that is, if is a right ideal of such that vanishes, then for a finite subset of . Clearly, a right zip ring is a right Beachy-Blair ring.

Ouyang et al. in [8] generalized the right Beachy-Blair condition from rings into modules as follows: A right -module is called module with the Beachy-Blair condition provided that if the right annihilator of a submodule of is zero, then there exists a finite subset such that .

The main aim of the present paper is to investigate conditions for the Malcev-Neumann series modules to satisfy the right Beachy-Blair condition. The proofs of our results obtained here are very similar to those obtained by Ouyang et al. in [8] and by Salem et al. in [9].

2. Generalized Malcev-Neumann Series Modules with the Beachy-Blair Condition

We start this section with the following notions and definitions.

Let be a subset of ; then

Definition 1. A ring is called -compatible if, for all and , if and only if .

Definition 2. A right -module is called -compatible if, for each , , and , if and only if .

Definition 3. A ring is called -Armendariz if whenever implies for each and , where and are elements of .

We extend the -Armendariz concept to modules as follows.

Definition 4. A right -module is called -Armendariz if whenever implies for each and , where and .

It is clear that is an -Armendariz (-compatible) ring if and only if is an -Armendariz (-compatible) module.

For a subset of , we define as the set

Lemma 5. Let be a right -module. Then , for any subset of .

Proof. Let . Then for each we have . Thus

which implies that for each . Hence for each . So and .
On the other hand, suppose that ; then for each . Thus for each , which implies that for each and . Hence and . So . Therefore .

When we have the following consequence of Lemma 5.

Corollary 6. Consider , for any subset of .

Note the following: for , let and for a subset , we have .

Lemma 7. Let be an -compatible and -Armendariz -module. Then

for any .

Proof. Let and . We show that and it is enough to show that for each . In fact, let . Then . Since is an -Armendariz module, for each and . Then for each . Thus and . Now, let . Then for each . Hence for each and . Since is -compatible, it follows that , which implies that for each and . Consequently

So and it follows that . So

For a right -module , we define

Lemma 5 gives us the map defined by for every . Obviously is an injective map.

In the following lemma we show that is a bijective map if and only if is -Armendariz.

Lemma 8. Let be an -compatible -module. The following conditions are equivalent.

(1) is an -Armendariz -module.(2) defined by is a bijective map.

Proof. (1)(2).
It is only necessary to show that is surjective. Let and . Since , the proof of this direction follows directly from Lemma 7.
(2)(1).
Let and such that . Then . By assumption for some right ideal of . Hence which implies that for each . So, and we have that

for each and . Thus for each and . So, is an -Armendariz module.

Recall that a ring is reduced if it has no nonzero nilpotent elements. Reduced rings have been studied for over forty-eight years (see [10]). In 2004, the reduced ring concept was extended to modules by Lee and Zhou [11] as follows: a right -module is reduced if, for any and any , implies . Clearly, if is reduced, then, for all and , implies . It is clear that is a reduced ring if and only if is a reduced module.

Now, we are able to prove the main result.

Theorem 9. Let be a reduced, -compatible, and -Armendariz right -module. If satisfies the right Beachy-Blair condition, then satisfies the right Beachy-Blair condition.

Proof. Suppose that a right -module satisfies the right Beachy-Blair condition and is a right -submodule of such that .
From Lemma 8, we conclude that . Thus .
Let denote the right -submodule of generated by . Since , we have . Since satisfies the right Beachy-Blair condition, there exists a finite subset

such that = . Let

Then is a finite subset of . Now we will see that . Let ; then for and. Since is a reduced -module, then for and. Then for each , we have . Therefore = , and so = is proved.
For each , there exists an element such that . Let be a minimal subset of such that for each ; then is a finite subset of and . Thus . Now we show that . Let the contrary; that is, , and suppose that ; then for each . Let ; since is an -Armendariz and -compatible module, we have for all and each . Hence , a contradiction. Hence is proved. Thus satisfies the right Beachy-Blair condition.

When we have the following consequence of Theorem 9.

Corollary 10. Suppose that is a reduced, -compatible, and -Armendariz ring. If satisfies the right Beachy-Blair condition, then satisfies the right Beachy-Blair condition.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author would like to express deep gratitude to the referee for his/her valuable suggestions which improved the presentation of the paper.

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