ABSTRACT

We introduce nil 3-Armendariz rings, which are generalization of 3-Armendariz rings and nil Armendaiz rings and investigate their properties. We show that a ring is nil 3-Armendariz ring if and only if for any n Î ℕ, is nil 3-Armendariz ring. Also we prove that a right Ore ring R is nil 3-Armendariz if and only if so is Q, where Q is the classical right quotient ring of. With the help of this result, we can show that a commutative ring is nil 3-Armendariz if and only if the total quotient ring of is nil 3-Armendariz.

1. Introduction

Throughout this article, denotes an associative ring, not necessary with identity. Given a ring the polynomial ring over is denoted by The study of Armendariz ring was initiated by Armendariz [1] and Rege and Chhawchharia [2]. A ring is called Armendariz if whenever polynomials satisfy then for all i, j. (The converse is always true.) Some properties of Armendariz rings have been studied in Rege and Chhawchharia [2], Anderson and Camillo [3], Kim and Lee [4], Huh et al. [5], and Lee and Wong [6]. Suiyi [7] introduced the notion of 3-Armendariz ring. A ring is called a 3-Armendariz if whenever polynomials satisfy then for all Due to Ramon Antoine [8], a ring is said to be nil Armendariz if whenever two polynomials satisfy then for all and There is a nil Armendariz ring but not Armendaiz by [8, Example 4.11]. A ring is called reduced if it has no nonzero nilpotent elements. Armendariz rings are thus a generalization of reduced rings, and therefore, nilpotent elements play an important role in this class of rings. There are many examples of rings with nilpotent elements which are Armendariz. In fact, in [3], Anderson and Camillo prove that if then is an Armendariz ring if and only if is reduced. In [9], Liu and Zhao introduced weak Armendariz rings as a generalization of Armendariz rings. A ring is weak Armendariz if whenever the product of two polynomials is zero then the product of their coefficients is nilpotent. In [10], Wu Hui-feng introduced the concept of weak 3-Armendariz ring as a generalization of 3-Armendariz rings and weak Armendariz ring and investigated their properties. A ring is weak 3-Armendariz if whenever the product of three polynomials is zero then the product of their coefficients is nilpotent. Motivated by results in Suiyi [7], Liu and Zhao [9], Antoine [8], Kim and Lee [4], Rege and Chhawchharia [2], and Wu Hui-feng [10,11], we investigate a generalization of nil Armendariz rings and 3-Armemdariz rings which we call nil 3-Armendariz rings.

2. Nil 3-Armendariz Rings

If is a ring, denotes the set of all nilpotent elements in and if denotes the subset of of the coefficients of

Condition (P): For all if then ( See [7])

Lemma 2.1. [11, Proposition 1]. If is a reduced ring, then satisfies the condition (P), but the converse is not true.

Lemma 2.2. [7, Theorem 1]. If a ring satisfies condition (P), then R is a 3-Armendariz ring.

Proposition 2.3. Let be a ring such that If then for all and

Proof. Observe that is reduced. By Lemma 2.1, satisfies condition (P) and by Lemma 2.2, is 3-Armendariz. Suppose Then, if we denote by the corresponding polynomials in Since is 3-Armendariz, for all and Hence is nil for all and

Wu Hui-feng gives the following generalization of 3-Armendariz rings.

Definition 2.4. [10, Definition 1]. A ring is said to be a weak 3-Armendariz ring if whenever polynomials satisfy then for all  and

Clearly, 3-Armendariz rings are weak 3-Armendariz. We now present here a stronger condition, given by the property obtained in Proposition 2.3.

Definition 2.5. A ring is said to be nil 3-Armendariz if whenever polynomials satisfy then for all and

Observe that if then by Proposition 2.3, is nil 3-Armendariz. More generally we obtain the following.

Proposition 2.6. Let be a ring that satisfies the condition (P), and a nil ideal. Then is nil 3-Armendariz if and only if is nil 3-Armendariz.

Proof. We denote Since is nil, then Hence if and only if And, if and then if and only if Therefore is nil 3-Armendariz if and only if is nil 3-Armendariz.

The next results can be proved by using the technique used in the proof of [8, Lemma 2.5, Lemma 2.6].

Lemma 2.7. Let be a nil 3-Armendariz ring and If such that then if for we have

Lemma 2.8. If is a 3-Armendariz ring then

Proposition 2.9. If is a 3-Armendariz ring then is nil 3-Armendariz.

Proof. Suppose be such that Since is 3-Armendariz, by Lemma 2.8, is nilpotent and there exists such that Hence, since is 3-Armendariz, for all and by choosing the corresponding coefficient in each polynomial, we have and thus, Therefore is nil 3-Armendariz.

Proposition 2.10. The class of nil 3-Armendariz rings is closed under finite direct products.

Proof. Let be the finite direct product of where is nil 3-Armendariz. Suppose for some polynomials where are elements of the product ring

. Set and Since then So and so Thus

in Since is nil 3-Armendariz, then we have Now, for each there exist positive integers

such that in the ring If we take then it is clear that Therefore This means that is nil 3-Armendariz.

Lemma 2.11. Let be a subring of If is nil 3-Armendariz. Then so is

Proof. Let be such that Then Since is nil 3-Armendariz, then i.e., This means that S is nil 3-Armendariz.

We denote by the ring consisting of all n-by-n upper triangular matrices over In [10, Theorem 1], showed that is a weak 3-Armendariz if and only if is a weak 3-Armendariz ring for all n Î ℕ. Here we have a similar results for nil 3-Armendariz rings.

Proposition 2.12. Let be a ring. The following conditions are equivalent:

1) is nil 3-Armendariz;

2) for any is nil 3-Armendariz.

Proof. (2)Þ(1) We note that any subring of nil 3-Armendariz rings is nil 3- Armendariz by Lemma 2.11. Thus if is nil 3-Armendariz ring, then so is (1)Þ(2) Let and

be elements of It is easy to see that there exists an isomorphism of rings define by:

Assume that Let

Then

corresponds a polynomial with coefficients in under the isomorphism Because and

we have

for

Since is nil 3-Armendariz, there exists such that for any and any Let Then

Thus, and so

This shows that is nil 3-Armendariz.

Corollary 2.13. If is a 3-Armendariz ring, then, for any is nil 3-Armendariz ring.

In [10, Corollary 1], it is shown that a ring is a weak 3-Armendariz ring if and only if is a weak 3-Armendariz ring, where is the ideal of generated by and is a positive integer. For nil 3-Armendariz rings, we have the following result.

Proposition 2.14. Let be a ring and any positive integer. Then is nil 3-Armendariz if and only if is nil 3-Armendariz, where is the ideal of generated by

Proof. As where

is a subring of If is nil 3-Armendariz, then, by Proposition 2.12, we have that is nil 3-Armendariz, and so is S. Thus, is nil 3-Armendariz. Conversely, if is nil 3-Armendariz, then as a subring of is nil 3- Armendariz too.

Corollary 2.15. A ring is nil 3-Armendariz if and only if the trivial extension is nil 3-Armendariz.

Proof. It follows from Proposition 2.12.

From Proposition 2.12, one may suspect that if is nil 3-Armendariz then every n-by-n full matrix ring over is nil 3-Armendariz, where But the following example erases the possibility.

Example 2.16. Let be a ring and let Let

be polynomials in Then But

is not nilpotent. Thus is not nil 3-Armendariz. Now we can give the example of nil 3-Armendariz rings which are not 3-Armendariz.

Example 2.17. Let be a nil 3-Armendariz ring. Then the ring

is not 3-Armendariz by [7, Example 4], for but is a nil 3-Armendariz ring by Proposition 2.12, because is a subring of

Proposition 2.18. Let be a ring and an idempotent of If is central in then the following statements are equivalent:

1) is nil 3-Armendariz;

2) and are nil 3-Armendariz.

Proof. (2)Þ(1). Is obvious since and are subrings of

(1)Þ(2). Note that as rings. Thus the result follows from Proposition 2.10.

In [5, Theorem 11], it was shown that if is a reduced ideal of such that is Armendariz, then is Armendariz. In [10, Proposition 4], it is shown that if is a weak 3-Armendariz ring, then so is where is a nilpotent ideal of We show that this result also holds for nil 3-Armendariz rings in the following.

Proposition 2.19. Let be a ring such that is a nil 3-Armendariz ring for some proper ideal of If then is nil 3-Armendariz.

Proof. Let such that Then

Since is nil 3-Armendariz, we have that Hence Since then This means that is a nil 3-Armendariz ring.

Anderson and Camillo in [3, Theorem 2], prove that a ring is Armendariz if and only if the polynomial ring is Armendariz. Yang Suiyi [7], prove that a ring is 3-Armendariz if and only if the polynomial ring is 3-Armendariz. In [11], it is shown that if is reduced ring, then and is 3-Armendariz ring. For nil 3-Armendariz rings we will give the following results.

Proposition 2.20. If is nil 3-Armendariz, then

Proof. Suppose and By Lemma 2.7, we have that where for In particular, for every is nilpotent. Therefore for all and hence

Theorem 2.21. If is a 3-Armendariz ring, then is a nil 3-Armendariz ring.

Proof. Let be 3-Armendariz ring. Then by [7, Theorem 3], is 3-Armendariz. Thus by Proposition 2.9, is nil 3-Armendariz.

Proposition 2.22. Let be a reduced ring. Then is a nil 3-Armendariz ring.

Proof. It follows from the method in the proof of [11, Theorem 1].

Corollary 2.23. If is a reduced ring, then is a nil 3-Armendariz ring.

Recall that an element of a ring is right regular if implies for Similarly, left regular elements can be defined. An element is regular if it is both left and right regular (and hence not a zero divisor).

A ring is called right (resp., left) Ore if given with regular, there exist with regular such that It is a well-known fact that is a right (resp., left) Ore ring if and only if the classical right (resp., left) quotient ring of exists.

Lemma 2.24. If then for any central element

Proof. Set Then Thus This means that

Theorem 2.25. Let be a right Ore ring with the classical right quotient ring If all right regular elements are central, then is nil 3-Armendariz if and only if so is

Proof. It suffices to show by Lemma 2.11, that if is nil 3-Armendariz rings so is We apply the proof of [5, Theorem 12]. Consider such that

By [12, Proposition 2.1.16], we can assume that with for all and a right regular elements Put

Then we have

Since by Lemma 2.24, Since is nil 3-Armendariz, for each and so for all Therefore is nil 3-Armendariz ring.

Corollary 2.26. Let be a ring and be a multiplicative closed subset in consisting of central regular elements. Then is nil 3-Armendariz rings if and only if is nil 3-Armendariz rings.

Corollary 2.27. A commutative ring is nil 3-Armendariz if and only if so is the total quotient ring of

Proof. It suffices to show the necessity by Lemma 2.11. Let be the multiplicative closed subset of all regular elements in . Then is the total quotient ring of and hence the result holds by Corollary 2.26.

The ring of Laurent polynomials in with coefficients in a ring consists of all formal sum

with obvious addition and multiplicationwhere and are (possibly negative) integers and denote it by

Corollary 2.28. Let be a ring. is nil 3- Armendariz if and only if is nil 3-Armendariz.

Proof. It suffices to establish necessity since is a subring of Let then clearly is a multiplicatively closed subset in consisting of central regular elements. Note that If is nil 3-Armendariz, so is by Corollary 2.26.

3. Acknowledgements

This paper is partially supported by National Natural Science Foundation of China (No.11261050). I also thank the referee for his or her valuable comments.

REFERENCES