Journal of Applied Mathematics

Volume 2013 (2013), Article ID 309392, 7 pages

http://dx.doi.org/10.1155/2013/309392

## -Skew -McCoy Rings

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 22 April 2013; Revised 3 July 2013; Accepted 10 July 2013

Academic Editor: Baolin Wang

Copyright © 2013 Areej M. Abduldaim and Sheng Chen. 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

As a generalization of -skew McCoy rings, we introduce the concept of -skew -McCoy rings, and we study the relationships with another two new generalizations, -skew -McCoy rings and -skew -McCoy rings, observing the relations with -skew McCoy rings, -McCoy rings, -skew Armendariz rings, -regular rings, and other kinds of rings. Also, we investigate conditions such that -skew -McCoy rings imply -skew -McCoy rings and -skew -McCoy rings. We show that in the case where is a nonreduced ring, if is 2-primal, then is an -skew -McCoy ring. And, let be a weak ()-compatible ring; if is an -skew -McCoy ring, then is -skew -McCoy.

#### 1. Introduction

Throughout this paper is an associative ring with identity, unless otherwise stated, and is an endomorphism of . The polynomial ring over with respect to is denoted by (or simply, the skew polynomial ring) which elements are polynomials in with coefficients in , the addition is defined as usual and the multiplication depending on the relation for any . Most of the results in the polynomial rings have been done with the case where is the identity. For a ring , is the prime radical (i.e., the intersection of all prime ideals of ), and is the set of all nilpotent elements of . A ring is said to be an ring if forms an ideal of .

Following Nielsen [1], a ring is said to be right McCoy; if two polynomials and such that , then there exists which satisfies . A left McCoy ring is defined similarly. If a ring is both right and left McCoy, then it is called a McCoy ring. Commutative rings are McCoy [2].

Başer et al. in [3] introduced the notion of -skew McCoy ring with respect to an endomorphism . Let be an endomorphism of a ring . The ring is called -skew McCoy; if two polynomials and such that , then for some *∖*. It is clear that a ring is right McCoy if is -skew McCoy, where is the identity endomorphism of .

Jeon et al. in [4] studied a generalization of McCoy rings, which they have been called -McCoy rings. A ring is said to be -McCoy; if whenever , then for some , where and are in . Thus the concept of -McCoy rings is a generalization of the concept of McCoy rings, but the converse may not be true in general.

There are many relationships between McCoyness and other kinds of rings like Armendariz rings, regular rings, reduced rings (i.e., a ring without nonzero nilpotent elements), 2-primal rings (i.e., if ), and others. An Armendariz ring is defined by Rege and Chhawchharia in [5]; if two polynomials and such that , then for all and . Also, it is proved in [5] that every Armendariz ring is McCoy, but the converse needs not to be true.

According to Hong et al. in [6, 7], the Armendariz property of a polynomial ring was extended to a skew polynomial ring. A ring is called -Armendariz (resp., -skew Armendariz); if two polynomials and such that , then (resp., ) for all and . Also, it is proved in [7] that any -Armendariz ring is -skew Armendariz, but the converse does not hold. Note that the notion of -skew McCoy rings extends both McCoy rings and -skew Armendariz rings [3].

Ouyang in [8] introduced the concept of a skew -Armendariz ring and showed that this notion generalizes the concept of -Armendariz ring defined by Hong et al. in [7]. Let be a ring with an endomorphism and an -derivation . The ring is called a skew -Armendariz ring; if two polynomials and such that , then for each and .

Motivated by all of the previous, in this paper, we introduced in Section 2 the concept of -skew -McCoy rings by considering the skew polynomial ring instead of the ring in the condition of -McCoy ring. Consequently, some results of -McCoy rings would be considered as special cases of -skew -McCoy rings. We showed that the notion of -skew -McCoy rings generalizes the notion of -skew McCoy rings introduced by Başer et al. [3]. We proved that, in case of commutative rings, if is a -regular ring but not regular, then is an -skew -McCoy ring. Also, if is a local, one-sided Artinian, nonreduced ring with an automorphism , then is an -skew -McCoy ring. Moreover, let be a nonreduced, right Noetherian ring. If is an Abelian -regular ring, then is an -skew -McCoy ring. In Section 3 we studied the relationships between -skew -McCoy rings and new two different concepts (-skew -McCoy ring and -skew -McCoy ring) that related and are close to the notion of -skew -McCoy rings and depending on various visions of -skew McCoy rings and -skew Armendariz rings. We proved that if is any ring, then is an -skew -McCoy ring for all . Also, let be an endomorphism of a ring , and Let be a semicommutative ring satisfies the -condition. If is an -skew -McCoy ring, then is -skew -McCoy. Furthermore, let be a weak (,)-compatible ring. If is an -skew -McCoy ring, then is -skew -McCoy.

Finally, we mention that skew polynomial rings play an important role and have applications in several domains like coding theory, Galois representations theory in positive characteristic, cryptography, control theory, and solving ordinary differential equations.

#### 2. -Skew -McCoy Rings

Motivating by [3, 4, 8], we introduced the following concept.

*Definition 1. *Let be an endomorphism of a ring . A ring is called -skew -McCoy; if two polynomials and such that , then , for some .

It is clear that every -McCoy ring is -skew -McCoy ring, where is the identity endomorphism of .

Proposition 2. *Every -skew McCoy ring is an -skew -McCoy ring. *

*Proof. *Suppose that is an -skew McCoy ring and for and . Then there exists a positive integer such that and , and there exists a positive integer such that and . Now, if and ( and , , and , , and ), then there exist such that and ( and , and , and and , resp.,) because is -skew McCoy, which implies that or for some . On the other hand, let , then . Since is -skew McCoy ring, hence and so is an -skew -McCoy ring.

The converse of Proposition 2 may not be true in general.

*Example 3. *Let be the ring of integers modulo . Consider the matrix ring over and an endomorphism defined by
For
we have , but for any , , thus is not -skew McCoy [3]. On the other hand is not -McCoy [4]. Furthermore, this shows that the upper triangular matrix ring
over is not -skew McCoy [3], but is a -McCoy ring [4]. In addition is an -skew -McCoy ring by Lemma 7 (d) below.

The idea of the following example appears in [4].

*Example 4. *Let be the full matrix ring over the power series ring over a field . Let
Let be the subring of generated by and . Let . Note that every element of is of the form
for some and . Let be an endomorphism of defined by
Consider two polynomials over ,
, but for every , , hence is not -skew -McCoy. On the other hand is not -McCoy [4].

*Example 5. *Let be the ring of integers modulo 4. Consider the ring
Let be an endomorphism defined by
is an -skew McCoy ring [3], hence is an -skew -McCoy ring.

*Example 6. *Consider the ring
where and are the set of all integers and all rational numbers, respectively. Let be an automorphism of defined by
is an -skew Armendariz ring [6], hence is -skew McCoy [3], and then is an -skew -McCoy ring.

Lemma 7. * (a) Let be a ring, and Let be an endomorphism of . If there exists a nonzero ideal of such that , then is -skew -McCoy. ** (b) Every nonsemiprime ring is an -skew -McCoy ring. ** (c) Let be a ring with at least one nonzero nilpotent ideal. Then is an -skew -McCoy ring. ** (d) Let be any ring. and are -skew -McCoy for . ** (e) Let be a ring, and Let be any positive integer; then is an -skew -McCoy ring, where is the ideal generated by . *

*Proof. * (a) Let . First, we assume that , then for all we have that . Secondly, let , hence for all nonzero , we obtain that . Thus is -skew -McCoy.

(b) Suppose that for a ring , and then , hence is an -skew -McCoy ring by (a).

(c) Since is nonsemiprime ring, then it is -skew -McCoy by (b).

(d) and are nonsemiprime rings, so they are -skew -McCoy by (b).

(e) Since is nonsemiprime ring, then is -skew -McCoy ring by (b).

*Remark 8. * The matrix ring over the semiprime ring considered in [4, Example 1.5] is -McCoy ring. By Lemma 7 if is non-smiprime, then is an -skew -McCoy ring. This is not necessary mean that “if is semiprime, then is not -skew -McCoy.” The matrix ring is also an -skew -McCoy ring by Lemma 7

A ring is called 2-primal by Birkenmeier et al. [9] if . Note that a 2-primal ring is reduced and a ring is 2-primal if and only if is reduced. It is easy to see that every reduced ring is -McCoy (which is unknown for -skew -McCoy ring), for this reason 2-primal rings are -McCoy, so we have the following for the case of an -skew -McCoy ring.

Proposition 9. *Let be a nonreduced ring. If is 2-primal, then is an -skew -McCoy ring. *

*Proof. *Assume that is 2-primal, and then , and since is nonreduced, then , hence is an -skew -McCoy ring by Lemma 7.

The converse of Proposition 9 needs not be true because the -skew -McCoy ring in Remark 8 is not 2-primal by [10].

Due to Jeon et al. [4], the class of -McCoy rings contains both McCoy rings and 2-primal rings. However, regular -McCoy rings are not McCoy or 2-primal [4]. Recall that a ring is -regular if there exist a positive integer and such that for every element . While is called a right (resp., left) -regular ring if there exists a positive integer and such that (resp., for every element , a ring is called strongly -regular if is both right and left -regular rings. It is known that every strongly -regular ring is -regular and every regular ring is -regular, but the converse may not be true. Also, note that is left -regular if it satisfies the DCC on chains of the form .

In the following example we show that -skew -McCoy ring may not be -regular.

*Example 10. *Let be the first Weyl algebra over a field of characteristic zero. Recall that , the polynomial ring with indeterminate and with . Now, let
where is not -regular and by [11], so that is nonsemiprime, and hence is an -skew -McCoy ring by Lemma 7. Furthermore, we have [11] which implies that is a 2-primal ring (which is nonreduced), and then is an -skew -McCoy ring by Proposition 9.

*Example 11. *If denotes the upper triangle matrix ring over a field, then is a -regular ring [12] and is an -skew -McCoy ring by Lemma 7.

In case that is a commutative ring, the concept of -regular rings coincides with the concept of strongly -regular rings. Also, every nonreduced ring is an -skew -McCoy ring. It is well known that if is commutative ring, then is regular if and only if is -regular and . In addition, every Artinian ring is -regular [13], so we have the following.

Proposition 12. *Let be a commutative ring. If is a -regular ring but not regular, then is -skew -McCoy. *

*Proof. *Since is a -regular ring but not regular, then , hence is -skew -McCoy by Lemma 7 *(b)*.

Corollary 13. *Let be a commutative not regular ring. If is Artinian, then is an -skew -McCoy ring. *

Corollary 14. *Let be a commutative ring. If is -regular, then is a -McCoy ring. *

Corollary 15. *If is commutative Artinian ring, then is -McCoy. *

Let be a ring, and Let be an endomorphism of ; Kwak in [14] defines an -ring to be a ring in which implies for . Also he called an ideal of a ring by completely prime if implies or for , .

Proposition 16. *Let be a nonreduced ring, and Let be an automorphism of . If is an -ring, then is -skew -McCoy. *

*Proof. *Since is an ()-ring, then by [15] is a 2-primal ring, therefore is -skew -McCoy by Proposition 9.

Corollary 17. *Let be a Noetherian nonreduced ring, and Let be an automorphism of . If for each minimal prime ideal of , and is completely prime ideal of , then is an -skew -McCoy ring.*

*Proof. *By [15] and Proposition 16.

Proposition 18. *If is a nonreduced, 2-primal ring with a nilpotent prime ideal, then is an -skew -McCoy ring.*

*Proof. *By [16] is a 2-primal ring, hence by Proposition 9 is an -skew -McCoy ring.

Chen [17] introduced the notion of semiabelian rings. A ring is semiabelian if where (i) is the set of idempotents in , (ii) (resp., ) is the set of right (resp., left) semicentral idempotents of , (iii) an idempotent in a ring is right (resp., left) semicentral if for every , (resp., ). Recall that a ring is Abelian if every idempotent element of is central and that a ring is right (resp., left) quasiduo if every maximal right (resp., left) ideal is an ideal, and a ring is quasiduo if it is right and left quasiduos.

Theorem 19. *Let be a right Noetherian ring. If is an Abelian -regular ring, then is 2-primal. *

*Proof. *Since is a right Noetherian ring, then every nil right or left ideal of is nilpotent [18], therefore contains all nil right or left ideals of , but is two sided [19] because is an Abelian -regular ring, hence which implies that is a 2-primal ring.

Badawi [19] and Chen [17] proved that if satisfies any one of the following: (a) an Abelian -regular ring; (b) a right (resp., left) quasiduo -regular ring; (c) a semiabelian -regular ring, then is an ideal of , so we have the following.

Corollary 20. *Let be a nonreduced right Noetherian ring. If is an Abelian -regular ring, then is -skew -McCoy. *

Corollary 21. *Let be a nonreduced, right Noetherian ring. If is a right (resp., left) quasiduo -regular ring, then is -skew -McCoy. *

Corollary 22. *Let be a nonreduced right Noetherian ring. If is a semiabelian -regular ring, then is -skew -McCoy. *

#### 3. Two Generalizations of -Skew McCoy Rings

As mentioned before that a ring with an endomorphism is called -skew McCoy ring; if two polynomials and such that , then for some [3]. In fact Song et al. in [20] introduced a concept of -skew McCoy rings in another way as a generalization of McCoy rings and -rigid rings (a ring with an endomorphism such that implies for in the ring). Let be an endomorphism of a ring , and let and with , is called a left -skew McCoy ring if there exists such that for all , and is called a right -skew McCoy ring if there exists such that for all . If a ring is both left -skew McCoy and right -skew McCoy, then is called an -skew McCoy ring. Every McCoy ring is an -skew McCoy ring, where is the identity endomorphism of . Here an -skew Armendariz ring may not be -skew McCoy in general [20], but if is an -skew Armendariz ring, then is right -skew McCoy [20].

As a generalization of the concept of -skew McCoy rings in the sense of Song et al. [20], we motivated by the previous to introduce the concepts of -skew -McCoy rings and -skew -McCoy rings taking into consideration the set of nilpotent elements of , . We gave examples to show that these two new concepts are not equivalent to each others and not equivalent to the concept of -skew -McCoy rings. Furthermore, we studied the relationship between each others as well as between them and -skew -McCoy rings on the other hand. We showed that if a certain property satisfies for -skew -McCoy rings may be this is not true for -skew ()-McCoy rings and vise versa. Also, we investigate some of their properties and characterizations.

*Definition 23. *Let be a ring, and Let be an endomorphism of . We say that is right -skew -McCoy; if two polynomials and with , then for any there exists (i.e., depending on ) such that . A left -skew -McCoy ring is defined similarly. If is both left and right -skew -McCoy, then is called an -skew -McCoy ring.

Every left -skew McCoy ring (in the sense of [20]) is left -skew -McCoy, but the converse may not be true in general. Also, if is a skew -Armendariz ring, then it is -skew -McCoy, so every -skew Armendariz ring is -skew -McCoy, again the converse needs not be true as in the following.

*Example 24. *Let and : be an endomorphism defined by . is not left -skew McCoy [20] and is not -skew Armendariz [6]. However, is an -skew -McCoy ring if , with , then it is clear that there exists such that for each .

We mention that there is no example of a ring which is not -skew -McCoy so far. However, it is convenient to show that the concept of -skew -McCoy rings and the concept of -skew -McCoy rings are not equivalent. In fact an -skew -McCoy ring may not be -skew -McCoy as we see in the following.

*Example 25. *Let be the subring as in Example 4 which is not -skew -McCoy. However, always we can find such that . So is an -skew -McCoy ring.

Proposition 26. *Let be any ring, and is an -skew -McCoy ring, for all . *

*Proof. *For any
take
then we have and , hence is an -skew -McCoy ring.

Recall that a ring is semicommutative if implies for , and that a ring is said to satisfy the -condition for an endomorphism of in case if and only if where [21]. In the following we show how may an -skew -McCoy ring imply -skew -McCoy.

Theorem 27. *Let be an endomorphism of a ring and be a semicommutative ring satisfies the -condition. If is an -skew -McCoy ring, then is -skew -McCoy. *

*Proof. *Let be an -skew -McCoy ring, and let , such that , and then there exists such that for each , hence . Since , then by [22], we have that for any positive integer , hence for each . Again by [22], we have , hence , so that is an -skew -McCoy ring.

Theorem 28. *Let be a Noetherian ring, and Let be an automorphism of which satisfies the -condition. If is an -skew -McCoy ring, then is -skew -McCoy. *

*Proof. *The proof is in the same steps of the proof of theorem 27 by using [22, Corollary 3.2] and [23, Proposition 2].

*Definition 29. *Let be an endomorphism of a ring . We say that is right -skew -McCoy; if two polynomials and with , then for any , there exists (i.e., depending on ) such that . A left -skew -McCoy ring is defined similarly. A ring is called -skew -McCoy if it is both left and right -skew -McCoy rings.

Every right -skew McCoy ring (in the sense of [20]) is right -skew -McCoy, but the converse may not be true in general, and likewise, every -skew Armendariz ring is -skew -McCoy, but the converse needs not be true as in the following.

*Example 30. *Let , and let be an endomorphism defined by . The ring is not -skew McCoy [20], and is not -skew Armendariz [6]. However is an -skew -McCoy ring; if , such that , then there exists such that for each .

Also here we mention that there is no example of a ring which is not -skew -McCoy so far.

*Remark 31. *As in the case of an -skew -McCoy ring, the concept of an -skew -McCoy ring is not equivalent to the concept of an -skew -McCoy ring, since the subring referred to in Example 25 is an -skew -McCoy ring because always we can find such that .

Ouyang [24] introduced the concept of weak (,)-compatible rings. For an endomorphism and -derivation , we say that is weak -compatible; if each , then if and only if . Moreover, is said to be weak -compatible; if each , , then . If is both weak -compatible and weak -compatible, then is said to be weak (,)-compatible. Now, it is clear that every -skew -McCoy ring is -skew -McCoy. In the following we show how we can make the converse true.

Proposition 32. *Let be a weak (,)-compatible ring. If is an -skew -McCoy ring, then is -skew -McCoy. *

*Proof. *Since is an -skew -McCoy ring, then for two polynomials and , , there exists such that , but is weak (,)-compatible ring, thus for every positive integer [24], hence for each , therefore is an -skew -McCoy ring.

Theorem 33. *Let be a weak (,)-compatible NI ring. If satisfies any one of the following: *(a)*is an -skew -McCoy ring;*(b)* is an -skew -McCoy ring; *(c)* is an -skew -McCoy ring; ** then is a skew -Armendariz ring. *

*Proof. *Let be any ring satisfies any one of (a), (b), and (c), hence for and with and by [24], we have for each , . Therefore is a skew -Armendariz ring.

Corollary 34. *Let be a weak ()-compatible, Abelian -regular ring. If satisfies any one of the following: *(a)* is an -skew -McCoy ring;*(b)* is an -skew -McCoy ring; *(c)* is an -skew -McCoy ring; ** then is a skew -Armendariz ring. *

Corollary 35. *Let be a weak ()-compatible, right (resp., left) quasiduo -regular ring. If satisfies any one of the following: *(a)* is an -skew -McCoy ring;*(b)* is an -skew -McCoy ring; *(c)* is an -skew -McCoy ring; ** then is a skew -Armendariz ring. *

Corollary 36. *Let be a weak ()-compatible, semiabelian -regular ring. If satisfies any one of the following: *(a)* is an -skew -McCoy ring;*(b)* is an -skew -McCoy ring; *(c)* is an -skew -McCoy ring; ** then is a skew -Armendariz ring. *

Theorem 37. *Let be a weak ()-compatible NI ring. If satisfies any one of the following: *(a)* is an -skew -McCoy ring;*(b)* is a skew -Armendariz ring; ** then is an -skew -McCoy ring. *

*Proof. *Let be any ring satisfies any one of (a), (b), and (c), hence for and with and by [24], there exists such that for all , therefore is an -skew -McCoy ring.

Corollary 38. *Let be a weak ()-compatible, Abelian -regular ring. If satisfies any one of the following: *(a)* is an -skew -McCoy ring;*(b)* is a skew -Armendariz ring; ** then is an -skew -McCoy ring. *

Corollary 39. *Let be a weak ()-compatible, right (resp., left) quasiduo -regular ring. If satisfies any one of the following: *(a)* is an -skew -McCoy ring;*(b)* is a skew -Armendariz ring; ** then is an -skew -McCoy ring. *

Corollary 40. *Let be a weak ()-compatible, semiabelian -regular ring. If satisfies any one of the following: *(a)* is an -skew -McCoy ring;*(b)* is a skew -Armendariz ring; ** then is an -skew -McCoy ring. *

#### Acknowledgments

Sheng Chen was supported by National Natural Science Foundation of China (Grant no. 11001064 and 11101105), by the Fundamental Research Funds for the Central Universities (Grant no. HIT. NSRIF. 2014085), and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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