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 ![](https://www.scirp.org/html/4-5300597\b82eab8c-662f-4764-9ffd-8aa331a9d1a9.jpg)
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 ![](https://www.scirp.org/html/4-5300597\17cf18c0-17ef-4afc-95ec-dc67be765ae4.jpg)
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 ![](https://www.scirp.org/html/4-5300597\4e9fbb94-a0a9-454f-80ab-e7966f899730.jpg)
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 ![](https://www.scirp.org/html/4-5300597\27fff9d1-6d7a-46f6-978e-b7300adde6eb.jpg)
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 ![](https://www.scirp.org/html/4-5300597\e99a6484-f38a-48be-94f9-6743ccdd727d.jpg)
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 ![](https://www.scirp.org/html/4-5300597\967264ff-dc2e-4e30-8038-0ce4d2fe5b5b.jpg)
Lemma 2.8. If
is a 3-Armendariz ring then ![](https://www.scirp.org/html/4-5300597\dac2f88f-377c-4b91-bcdf-dd0133fa2360.jpg)
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 ![](https://www.scirp.org/html/4-5300597\d40a1c3a-0b32-43c0-b8ad-322b1381cc09.jpg)
Thus
in
Since
is nil 3-Armendariz, then we have
Now, for each
there exist positive integers ![](https://www.scirp.org/html/4-5300597\42a28121-01c8-424f-80a9-9b5a66b8d591.jpg)
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 ![](https://www.scirp.org/html/4-5300597\5f6fb9f3-8d11-433a-8d71-72f8d0090eb9.jpg)
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:
![](https://www.scirp.org/html/4-5300597\d7d4710e-2b25-4df6-b9fe-e506a6ea64aa.jpg)
Assume that
Let
![](https://www.scirp.org/html/4-5300597\52ddd989-ae28-4787-bcfc-88baa4d54186.jpg)
Then
![](https://www.scirp.org/html/4-5300597\b7c56bee-31a9-4ec5-9273-9382993e1a39.jpg)
corresponds a polynomial with coefficients in
under the isomorphism
Because
and
![](https://www.scirp.org/html/4-5300597\0a940eca-a530-4dc4-9027-3878630f5b0d.jpg)
we have
![](https://www.scirp.org/html/4-5300597\2051b1ba-4338-4e33-980b-dd2cbacfb208.jpg)
for ![](https://www.scirp.org/html/4-5300597\5bed3d2e-56fa-494e-80ab-5b811d54cc53.jpg)
Since
is nil 3-Armendariz, there exists
such that
for any
and any
Let
Then
![](https://www.scirp.org/html/4-5300597\8b5c6655-a65f-44e9-8445-1156b89e704c.jpg)
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 ![](https://www.scirp.org/html/4-5300597\9b9d351b-624d-46c0-8dcb-09b9040b2df8.jpg)
Proof. As
where
![](https://www.scirp.org/html/4-5300597\99115a4a-a886-4330-bc4f-0bdb50a338e9.jpg)
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
![](https://www.scirp.org/html/4-5300597\79409be7-378e-4f25-b61c-f8263a7669ba.jpg)
be polynomials in
Then
But
![](https://www.scirp.org/html/4-5300597\fe9f2735-2d90-47c8-9e7a-c7bbbe932142.jpg)
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
![](https://www.scirp.org/html/4-5300597\250b82ca-04b2-434b-9b9c-a4ad606de8c8.jpg)
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 ![](https://www.scirp.org/html/4-5300597\b8af72d1-01e5-4f56-b07f-9b614eb1cbb9.jpg)
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 ![](https://www.scirp.org/html/4-5300597\75cfa970-2b60-4779-957c-d39f7052cb15.jpg)
(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 ![](https://www.scirp.org/html/4-5300597\b35b01b5-7544-4739-84cd-268778338470.jpg)
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 ![](https://www.scirp.org/html/4-5300597\a6f619a7-3c75-4e60-86d1-8fd653fa08a0.jpg)
Proof. Suppose
and
By Lemma 2.7, we have that
where
for
In particular, for every
is nilpotent. Therefore
for all
and hence ![](https://www.scirp.org/html/4-5300597\d75b3937-b12d-49d4-8279-45656c537381.jpg)
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
![](https://www.scirp.org/html/4-5300597\2617a3fc-7bed-4841-8c22-c800b5642b87.jpg)
Proof. Set
Then
Thus
This means that ![](https://www.scirp.org/html/4-5300597\6c4cac7a-386f-43c0-a49b-2d8c8f8b35e2.jpg)
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 ![](https://www.scirp.org/html/4-5300597\a854d03c-97e5-48a7-af77-632264bb6999.jpg)
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
![](https://www.scirp.org/html/4-5300597\219e05d5-7970-4c12-a159-4db05b82580a.jpg)
such that
By [12, Proposition 2.1.16], we can assume that
![](https://www.scirp.org/html/4-5300597\7afc44bd-7f85-4681-9f6e-fee704085c3d.jpg)
with
for all
and a right regular elements
Put
![](https://www.scirp.org/html/4-5300597\1cc45ddf-95dc-4cf1-81d8-42b1e0f6a527.jpg)
Then we have
![](https://www.scirp.org/html/4-5300597\fa03d849-e532-465f-838c-77584fe1cf12.jpg)
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 ![](https://www.scirp.org/html/4-5300597\2a8581c6-dd43-4a53-98fb-eeaa87702634.jpg)
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 ![](https://www.scirp.org/html/4-5300597\255b6ca3-1877-4efb-9f43-86d3a0460cdb.jpg)
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.