PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 95 (109) (2014), 221–228 DOI: 10.2298/PIM1409221N

ON WEAK α-SKEW MCCOY RINGS

Mohammad Javad Nikmehr, Ali Nejati and Mansoureh Deldar

Communicated by Žarko Mijajlović

Abstract. Let α be an endomorphism of a ring R. We introduce the notion of weak α-skew McCoy rings which are a generalization of the α-skew McCoy rings and the weak McCo rings. Some properties of this generalization are established, and connections of properties of a weak α-skew McCoy ring R with n × n upper triangular T_{n} (R) are investigated. We study relationship between the weak skew McCoy property of a ring R and its polynomial ring, R[x]. Among applications, we show a number of interesting properties of a weak α-skew McCoy ring R such as weak skew McCoy property in a ring R.

1. Introduction

Throughout this note, R denotes an associative ring with unity and α is a ring endomorphism. We denote R[x; α] the Ore extension whose elements are the polynomials ∑ _{i=0}^{n} a_{i}x^{i}, a_{i} ∈ R, where the addition is defined as usual and the multiplication subject to the relation xa = α(a)x for any a ∈ R. nil(R) denotes the set of all the nilpotent elements of R. Rege and Chhawchharia [7] introduced the notion of an Armendariz ring. They defined a ring R to be an Armendariz ring if whenever polynomials f(x) = a_{0}+a_{1}x+⋯ +a_{n}x^{n}, g(x) = b_{0}+b_{1}x+⋯ +b_{m}x^{m} ∈ R[x] satisfy f(x)g(x) = 0, then a_{i}b_{j} = 0 for each i, j. The name “Armendariz ring" was chosen because Armendariz had showed that a reduced ring (i.e., a ring without nonzero nilpotent elements) satisfies this condition. Hong, Kim, and Kwak [3] called R an α-skew Armendariz ring if whenever polynomials f(x) = a_{0}+a_{1}x+⋯ +a_{n}x^{n}, g(x) = b_{0} + b_{1}x + ⋯+ b_{m}x^{m} ∈ R[x; α] satisfy f(x)g(x) = 0, then a_{i}α^{i}(b_{j}) = 0 for each i, j, which is a generalization of the Armendariz rings. Liu and Zhao [4] called a ring R weak Armendariz if whenever polynomials f(x) = a_{0} + a_{1}x + ⋯+ a_{n}x^{n}, g(x) = b_{0} + b_{1}x + ⋯+ b_{m}x^{m} ∈ R[x] satisfy f(x)g(x) = 0, then a_{i}b_{j} is nilpotent element of R for each i and j. Motivated by the above results, Zhang and Chen [8] called a ring R weak α-skew Armendariz if whenever polynomials f(x) = a_{0} + a_{1}x + ⋯+ a_{n}x^{n}, g(x) = b_{0} + b_{1}x + ⋯+ b_{m}x^{m} ∈ R[x; α] satisfy f(x)g(x) = 0, then a_{i}α^{i}(b_{j}) ∈ nil(R) for each i and j. It is obvious that a weak α-skew Armendariz ring is a generalization of the α-skew Armendariz rings and the weak Armendariz rings. Recall that a ring R is called reversible if ab = 0 implies ba = 0, for all a, b ∈ R. R is called semicommutative if for all a, b ∈ R, ab = 0 implies aRb = {0}. Reduced rings are clearly reversible and reversible rings are semicommutative, but the converse is not true in general [6]. According to Nielson [6], a ring R is called right McCoy (resp., left McCoy) if, for any polynomials f(x), g(x) ∈ R[x] ∖ {0}, f(x)g(x) = 0 implies f(x)r = 0 (resp., sg(x) = 0) for some 0 ≠ r ∈ R (resp., for some 0 ≠ s ∈ R). A ring is called McCoy if it is both left and right McCoy. By McCoy [5], commutative rings are McCoy rings. Reduced rings are Armendariz and Armendariz rings are McCoy. A ring R is right weak McCoy whenever, f(x) = a_{0} + a_{1}x + ⋯ + a_{n}x^{n}, g(x) = b_{0} + b_{1}x + ⋯ + b_{m}x^{m} ∈ R[x] ∖ {0} satisfy f(x)g(x) = 0, then a_{i}s ∈ nil(R) for some 0 ≠ s ∈ R, and every i. Left weak McCoy rings are defined similarly. If a ring is both left and right weak McCoy we say that the ring is weak McCoy ring. Also in [2] investigated this generalization of McCoy rings and their properties.

2010 Mathematics Subject Classification: Primary 16S36; Secondary 16S50.

Key words and phrases: McCoy rings, skew polynomial rings, reduced rings.

A ring R is called α-skew McCoy ring with respect to α if for any nonzero polynomials f(x) = a_{0} + a_{1}x + ⋯+ a_{n}x^{n}, g(x) = b_{0} + b_{1}x + ⋯+ b_{m}x^{m} ∈ R[x; α] satisfy f(x)g(x) = 0, implies f(x)s = 0 for some nonzero s ∈ R. It is clear that a ring R is right McCoy if R is id_{R}-skew McCoy, where id_{R} is the identity endomorphism of R. In [1], Basser, Kwak, Lee showed that every domain with an endomorphism α is α-skew McCoy, and R is α-skew McCoy if and only if the factor ring R[x]/(x^{n}) is ŕᾱ-skew McCoy, where ᾱ : R[x] → R[x] defined by ᾱ(f(x)) = ∑ _{i=0}^{m} α(a_{i})x^{i} for any f(x) = a_{0} + a_{1}x + ⋯+ a_{n}x^{n} is an endomorphism of R[x]. Also they proved that for a ring isomorphism σ : R → S, R is a α-skew McCoy ring if and only if S is an σασ^{−1} -skew McCoy ring.

Motivated by the above results, for an endomorphism α of a ring R, we investigate a generalization of the α-skew McCoy rings and the weak McCoy rings which we call a weak α-skew McCoy ring and study several results.

2. Weak α-Skew McCoy rings

We begin this section by the following definition and also we study properties of weak α-skew McCoy rings.

Definition 2.1. Let α be an endomorphism of a ring R. The ring R is called weak α-skew McCoy with respect to α if for any nonzero polynomials p(x) = ∑ _{i=0}^{n} a_{i}x^{i} and q(x) = ∑ _{j=0}^{m} b_{j}x^{j} in R[x; α] with p(x)q(x) = 0, there exists s ∈ R − {0} such that a_{i}α^{i}(s) ∈ nil(R) for 0 ⩽ i ⩽ n.

It can be easily checked that if R is a weak McCoy ring then it is a weak id_{R}-skew McCoy ring, where id_{R} is an identity endomorphism of R. Also every weak Armendariz ring is weak McCoy and therefore is weak id_{R}-skew McCoy. If nil(R) ⊴ R, then R is weak Armendariz and so R will be weak McCoy ring and so R is weak id_{R}-skew McCoy.

ON WEAK α-SKEW MCCOY RINGS 223

Proposition 2.1. Let α be an endomorphism of a ring R. Then every weak α-skew Armendariz ring is a weak α-skew McCoy ring.

Proof. Let f(x) = ∑ _{i=0}^{n} a_{i}x^{i}, g(x) = ∑ _{j=0}^{m} b_{j}x^{j} ∈ R[x; α] ∖ {0} and assume that f(x)g(x) = 0. Since R is weak α-skew Armendariz, a_{i}α^{i}(b_{j}) ∈ nil(R) for all i, j. Let r = b_{t} for 0 ⩽ t ⩽ m, and hence a_{i}α^{i}(r) ∈ nil(R) for all i. Therefore R is weak α-skew McCoy. □

Let I be an ideal of R. If α(I) ⊆ I, then defined ᾱ : R/I → R/I by ᾱ(a + I) =

α(a) + I for a ∈ R, is an endomorphism of the factor ring R/I. Now we have the following proposition.

Proposition 2.2. Let α be an endomorphism of a ring R and I be an ideal of R with α(I) ⊆ I. If I ⊆ nil(R) and R/I is weak ᾱ-skew McCoy, then R is weak α-skew McCoy.

Proof. Let f(x) = a_{0} + a_{1}x + ⋯+ a_{m}x^{m} and g(x) = b_{0} + b_{1}x + ⋯+ b_{n}x^{n} ∈

R[x; α] ∖ {0} such that f(x)g(x) = 0. Then ( ∑ _{i=0}^{m} ā_{i}x^{i})( ∑ _{j=0}^{n} b̄_{j}x^{j} ) = 0 in R/I. Thus there exists n_{i} such that (ā_{i} ᾱ^{i} (s̄))^{n_{i}} = 0 for some s ∈ R ∖ I. Hence a_{i}α^{i}(s) ∈ nil(R) and so R is weak α-skew McCoy. □

Let R be a ring, α an automorphism of R and Δ a multiplicatively closed subset of R consisting of central regular elements. The ring Δ^{−1} R is called the ring of fractions of R with respect to Δ. We define Δ^{−1} α : Δ^{−1} R → Δ^{−1} R by Δ^{−1} α(b^{−1} a) = (α(b))^{−1} α(a) for any b^{−1} a ∈ Δ^{−1} R. Then Δ^{−1} α is an automorphism of Δ^{−1} R.

Proposition 2.3. Let R be weak α-skew McCoy. Then Δ^{−1} R is weak Δ^{−1} αskew McCoy.

Proof. Let f(x) = ∑ _{i=0}^{m} c_{i}x^{i} and g(x) = ∑ _{j=0}^{n} d_{j}x^{j} be nonzero polynomials in Δ^{−1} R[x; Δ^{−1} α] such that c_{i}, d_{j} are in Δ^{−1} R for all i, j. Then we can assume that c_{i} = a_{i}u^{−1} and d_{j} = b_{j} v^{−1} for some a_{i}, b_{j} ∈ R and u, v ∈ Δ. Let f_{1}(x) = ∑ _{i=0}^{m} a_{i}x^{i}, g_{1}(x) = ∑ _{j=0}^{n} b_{j}x^{j} . Thus f_{1}(x)g_{1}(x) = 0 inR[x; α]. Thus a_{i}α^{i}(s) ∈ nil(R) for some 0 ≠ s ∈ R for 0 ⩽ i ⩽ m. So c_{i}(Δ^{−1} α)^{i}(s) ∈ nil(Δ^{−1} R) for 0 ⩽ i ⩽ m. Thus Δ^{−1} R is a weak Δ^{−1} α-skew McCoy ring. □

Let R[x; x^{−1} ] be the ring of Laurent polynomials, i.e., the formal sums ∑ _{i=k}^{n} a_{i}x^{i}, where k, n are (possibly negative) integers. For an automorphism α of R, ᾱ : R[x; x^{−1} ] → R[x; x^{−1} ] defined by ᾱ( ∑ _{i=k}^{n} a_{i}x^{i})=∑ _{i=k}^{n} α(a_{i})x^{i} is an automorphism of R[x; x^{−1} ]. The restriction of ᾱ to R[x], we also denote by ᾱ.

Corollary 2.1. Let R[x] be weak ᾱ-skew McCoy ring. Then R[x; x^{−1} ] is a weak ᾱ-skew McCoy ring.

Proof. It is clear that Δ = {1, x, x^{2}, …} is multiplicatively closed subset of R[x]. Since R[x; x^{−1} ] = Δ^{−1} R[x], it follows that R[x; x^{−1} ] is a weak ᾱ-skew McCoy ring. □

Let α be an endomorphism of a ring R and M_{n}(R) be the n × n matrix over R, and ᾱ : M_{n}(R) → M_{n}(R) defined by ᾱ((a_{ij} )) = (α(a_{ij} )). Then ᾱ is an endomorphism of M_{n}(R). It is obvious that the restriction of ᾱ to T_{n}(R) is an endomorphism of T_{n}(R), where T_{n}(R) is the n × n upper triangular matrix ring over R. We also denote ᾱ|_{T_{n} (R)} by ᾱ.

For a ring R, T_{n}(R) (n ⩾ 2) is a weak McCoy ring. Now we have the following proposition.

Proposition 2.4. Let α be an endomorphism of a ring R. Then, for any n, T_{n}(R) is a weak ᾱ-skew McCoy ring if R is a weak α-skew McCoy ring.

Proof. Let f(x) = A_{0} + A_{1}x + ⋯+ A_{p}x^{p} and g(x) = B_{0} + B_{1}x + ⋯+ B_{q}x^{q}

be elements of T_{n}(R)[x; ᾱ] satisfying f(x)g(x) = 0, where

A_{i} = ( \begin{matrix}a_{11}^{(i)} & a_{12}^{(i)} & ⋯ & a_{1n}^{(i)} \\ 0 & a_{22}^{(i)} & ⋯ & a_{2n}^{(i)} \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & a_{nn}^{(i)} \end{matrix} ) , B_{j} = ( \begin{matrix}b_{11}^{(j)} & b_{12}^{(j)} & ⋯ & b_{1n}^{(j)} \\ 0 & b_{22}^{(j)} & ⋯ & b_{2n}^{(j)} \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & b_{nn}^{(j)} \end{matrix} ) .

Then from f(x)g(x) = 0, it follows that (∑ _{i=0}^{p} a_{ss}^{(i)} x^{i} )(∑ _{j=0}^{q} b_{ss}^{(j)} x^{j}) = 0 in R[x; α] for each s with 1 ⩽ s ⩽ n. Since R is a weak α-skew McCoy ring, there exists s_{k} ≠ 0 such that a_{ss}^{(i)} α^{i}(s_{k}) ∈ nil(R) for 1 ⩽ k ⩽ n. Therefore (a_{ss}^{(i)} α^{i}(s_{k}))^{m_{k}} = 0 for some m_{k} ∈ ℤ. Let m = max{m_{1}, m_{2}, …, m_{n}}. We define

S = ( \begin{matrix}s_{1} & ∗ & ⋯ & ∗ \\ 0 & s_{2} & ⋯ & ∗ \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & s_{n} \end{matrix} ) ,

where ∗ stands for any element of R. Then

(A_{i} ᾱ^{i} (S))^{m} = ( \begin{matrix}a_{11}^{(i)} α^{i}(s_{1}) & ∗ & ⋯ & ∗ \\ 0 & a_{22}^{(i)} α^{i}(s_{2}) & ⋯ & ∗ \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & a_{nn}^{(i)} α^{i}(s_{n}) \end{matrix} )^{m} = ( \begin{matrix}0 & ∗ & ⋯ & ∗ \\ 0 & 0 & ⋯ & ∗ \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & 0 \end{matrix}) .

It implies that T_{n}(R) is a weak ᾱ-skew McCoy ring. □

Example 2.1. [1] Let α be an endomorphism on the 2 × 2 matrices ring R =

M_{2}(ℤ_{3}) over ℤ_{3} defined by α(( \begin{matrix}a & b \\ c & d \end{matrix})) = ( \begin{matrix}a & −b \\ −c & d \end{matrix} ). For p(x) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) + ( \begin{matrix}1 & 1 \\ 0 & 0 \end{matrix} )x,

q(x) = ( \begin{matrix}0 & 0 \\ 0 & −1 \end{matrix} ) + ( \begin{matrix}0 & 1 \\ 0 & 1 \end{matrix} )x ∈ R[x; α], one has p(x)q(x) = 0. It can be easily checked that p(x)c ≠ 0 for any nonzero c ∈ R. Therefore R is not α-skew McCoy. This also shows that the 2 × 2 upper triangular matrix ring \{( \begin{matrix}a & b \\ 0 & c \end{matrix}) | a, b, c ∈ ℤ_{3}\} over ℤ_{3} is not α-skew McCoy.

We note that the α-skew McCoy ring is weak α-skew McCoy, but the converse is not always true by the following example.

Example 2.2. Since R = ℤ_{3} is a domain, it is α-skew Armndariz ring for any endomorphism α of R by [3, Proposition 10]. Hence R is α-skew McCoy. Thus R is weak α-skew McCoy, therefore T_{2}(ℤ_{3}) is weak ᾱ-skew McCoy ring by Propositin 2.4. But T_{2}(ℤ_{3}) is not α-skew McCoy ring the Example 2.1.

In the following, we provide a connection between abelian and weak α-skew McCoy rings.

Proposition 2.5. Let R be an abelian ring and α be an endomorphism with α(e) = e for every e^{2} = e ∈ R. Then R is a weak α-skew McCoy ring if eR and (1 − e)R are weak α-skew McCoy for some e^{2} = e ∈ R.

Proof. Let f(x) = a_{0} + a_{1}x + ⋯+ a_{m}x^{m}, g(x) = b_{0} + b_{1}x + ⋯+ b_{n}x^{n} in

R[x; α] with f(x)g(x) = 0. Let f_{1}(x) = ef(x), f_{2}(x) = (1 − e)f(x), g_{1}(x) = eg(x), g_{2}(x) = (1−e)g(x). Then f_{1}g_{1}(x) = 0, f_{2}g_{2}(x) = 0. Since eR and (1−e)R are weak α-skew McCoy, there exist m_{i}, n_{i} such that e(a_{i}α^{i}(s))^{m_{i}} = ((ea_{i})α^{i}(es))^{m_{i}} = 0 and (1 − e)(a_{i}α^{i}(t))^{n_{i}} = (((1 − e)a_{i}) α^{i}((1 − e)t)) ^{n_{i}} = 0 for some s ∈ eR, t ∈ (1 − e)R. Let k_{i} = max{m_{i}, n_{i}}. Then (a_{i}α^{i}(st))^{k_{i}} = 0. This means that R is weak α-skew McCoy. □

Let R_{i} be a ring and α_{i} an endomorphism of R_{i} for each i ∈ I. Then, for the product ∏ _{i∈I} R_{i} of R_{i} and the endomorphism ᾱ : ∏ _{i∈I} R_{i} → ∏ _{i∈I} R_{i} defined by ᾱ((a_{i})) = (α_{i}(a_{i})), ∏ _{i∈I} R_{i} is weak ᾱ-skew McCoy if and only if each R_{i} is weak α_{i}-skew McCoy.

Every homomorphism σ of rings R and S can be extended to the homomorphism of rings R[x] and S[x] defined by ∑ _{i=0}^{m} a_{i}x^{i} ↦ ∑ _{i=0}^{m} σ(a_{i})x^{i}, which we also denote by σ.

Proposition 2.6. Let σ : R → S be a ring isomorphism. If R is weak α-skew McCoy, then S is weak σασ^{−1} -skew McCoy.

Proof. Assume that f(x) = ∑ _{i=0}^{m} a_{i}x^{i} and g(x) = ∑ _{j=0}^{m} b_{j}x^{j} are polynomials in S[x, σασ^{−1} ]. Since σ is an isomorphism, there exist f_{1}(x) = ∑ _{i=0}^{m} a′_{i}x^{i} and g(x) = ∑ _{j=0}^{m} b′_{j}x^{j} in R[x, α] such that f(x) = σ(f_{1}(x)) = ∑ _{i=0}^{m} σ(a′_{i})x^{i} and g(x) = σ(g_{1}(x)) = ∑ _{j=0}^{m} σ(b′_{j})x^{j} . First we show that f(x)g(x) = 0 implies f_{1}(x)g_{1}(x) = 0. We have

a_{0}b_{k} + a_{1}(σασ^{−1} )(b_{k−1} ) + ⋯+ a_{k}(σασ^{−1} )^{k}(b_{0}) = 0 for any 0 ⩽ k ⩽ m.

From the definition of f_{1}(x) and g_{1}(x), we have,

σ(a′_{0})σ(b′_{k}) + σ(a′_{1})(σασ^{−1} )σ(b′_{k−1} ) + ⋯+ σ(a′_{k})(σασ^{−1} )^{k}σ(b′_{0}) = 0,

so that (σασ^{−1} )^{t} = σα^{t}σ^{−1} we obtain a′_{0}b′_{k} +a′_{1}α(b′_{k−1} )+⋯ +a′_{k}α^{k}(b′_{0}) = 0, which means that f_{1}(x)g_{1}(x) in R[x; α]. From the fact that R is weak α-skew McCoy, we have a′_{i}α^{i}(r) ∈ nil(R) for some r ∈ R. Since a′_{i} = σ^{−1} (a_{i}), r = σ^{−1} (s) for some s ∈ S, we have σ^{−1} (a_{i})α^{i}(σ^{−1} s) ∈ nil(R). Therefore we obtain a_{i}(σασ^{−1} )^{i}(s) ∈ nil(R), 0 ⩽ i, j ⩽ m. Hence S is weak σασ^{−1} -skew McCoy. □

Let E_{ij} = (e_{st}), 1 ⩽ s, t ⩽ n, denotes n × n unit matrices over ring R, in which e_{ij} = 1 and e_{st} = 0 when s ≠ i or t ≠ j, 0 ⩽ i, j ⩽ n for all n ⩾ 2. If V = ∑ _{i=1}^{n−1} E_{i,i+1} , then V_{n}(R) = RI_{n} + RV + ⋯+ RV^{n−1} is the subring of upper triangular skew matrices.

Corollary 2.2. Suppose that α is an endomorphism of a ring R. If the factor ring \frac{R[x]}{(x^{n} )} is weak ᾱ-skew McCoy, then V_{n}(R) is weak ᾱ-skew McCoy.

Proof. Assume that R[x]/(x^{n}) is weak ᾱ-skew McCoy and define the ring isomorphism θ : V_{n}(R) → R[x]/(x^{n}) defined by

θ(r_{0}I_{n} + r_{1}V + ⋯+ r_{n−1} V^{n−1} ) = r_{0} + r_{1}x + ⋯+ r_{n−1} x^{n−1} + (x^{n}).

Now we have that V_{n}(R) is weak θ^{−1} ᾱθ-skew McCoy and that

θ^{−1} ᾱθ(r_{0}I_{n} + r_{1}V + ⋯+ r_{n−1} V^{n−1} ) = ᾱ(r_{0}I_{n} + r_{1}V + ⋯+ r_{n−1} V^{n−1} ),

which means that V_{n}(R) is a weak ᾱ-skew McCoy ring. □

Before stating Theorem 2.1, we need the following proposition.

Proposition 2.7. [8] Let R be a reversible ring and α be an endomorphism of R such that aα(b) = 0 whenever ab = 0 for any a, b ∈ R. Then R is weak α-skew Armendariz.

In [4] it was shown that if a ring R is semicommutative, then R[x] is weak Armendariz. For the case of weak α-skew McCoy, we have the following theorem.

Theorem 2.1. Let R be a reversible ring and α be an endomorphism of R such that aα(b) = 0 whenever ab = 0 for any a, b ∈ R. If for some positive integer t, α^{t} = 1_{R}, then R[x] is weak α-skew McCoy.

Proof. Let p(y) = f_{0}(x)+f_{1}(x)y +⋯ +f_{m}(x)y^{m} and q(y) = g_{0}(x)+g_{1}(x)y +

⋯+ g_{n}(x)y^{n} be in (R[x])[y; α] with p(y)q(y) = 0. We also let f_{i}(x) = a_{i0} + a_{i1}x + ⋯+ a_{iw_{i}} x^{w_{i}} and g_{j} (x) = b_{j0} + b_{j1} x + ⋯+ b_{jv_{j}} x^{v_{j}} for any 0 ⩽ i ⩽ m and 0 ⩽ j ⩽ n, where a_{i0}, a_{i1}, …, a_{iw_{i}} , b_{j0} , b_{j1} , …, b_{jv_{j}} ∈ R. Take a positive integer k such that k > deg(f_{0}(x))+deg(f_{1}(x))+⋯ +deg(f_{m}(x))+deg(g_{0}(x))+deg(g_{1}(x))+ ⋯+ deg(g_{n}(x)), where the degrees of f_{i}(x) and g_{j} (x) are as the polynomials in R[x] and the degree of zero polynomial is taken to be 0 for all 0 ⩽ i ⩽ m and 0 ⩽ j ⩽ n. Let f(x) = f_{0}(x^{t}) + f_{1}(x^{t})x^{tk+1} + f_{2}(x^{t})x^{2tk+2} + ⋯+ f_{m}(x^{t})x^{mtk+m} and g(x) = g_{0}(x^{t}) + g_{1}(x^{t})x^{tk+1} + g_{2}(x^{t})x^{2tk+2} + ⋯+ g_{n}(x^{t})x^{ntk+n} ∈ R[x]. Then the set of coefficients of the f_{i}(x) (respectively, g_{j} (x)) equals the set of coefficients of f(x) (respectively, g(x)). Since p(y)q(y) = 0, x commutes with elements of R in the polynomial ring R[x], and α^{t} = 1_{R}, we have f(x)g(x) = 0 in R[x; α]. By Proposition 2.7, R is weak α-skew Armendariz, and so R weak α-skew McCoy by Proposition 2.1. Thus there exists b ≠ 0 in R such that a_{il} α^{i}(b) ∈ nil(R) for any 0 ⩽ i ⩽ m, l ∈ {0, 1, …, w_{0}, …, w_{m}}. Since R is reversible, ∑ _{l} a_{il} α^{i}(b) ∈ nil(R), by [4, Lemma 3.1]. Therefore f_{i}(x)α^{i}(b) ∈ nil(R[x]) by [4, Lemma 3.7] for all i, and hence R[x] is weak ᾱ-skew McCoy. □

Also, for the weak α-skew McCoy, the following result holds.

Theorem 2.2. Let R be a reversible ring and α be an endomorphism of R such that aα(b) = 0 whenever ab = 0 for any a, b ∈ R. If, for some positive integer t, α^{t} = 1_{R}, then R[x; α] is weak α-skew McCoy.

Proof. Let p(y), q(y) and k be the same as in the proof of Theorem 2.1. We claim that f_{i}(x)g_{j} (x) ∈ nil(R[x; α]) for all 0 ⩽ i ⩽ m, 0 ⩽ j ⩽ n. Let p(x^{tk}) = f_{0}(x)+f_{1}(x)x^{tk} +⋯ +f_{m}(x)x^{mtk} and q(x^{tk}) = g_{0}(x)+g_{1}(x)x^{tk} +⋯ +g_{n}(x)x^{ntk} ∈ R[x; α]. Then the set of coefficients of f_{i}(x) (respectively, g_{j} (x)) equals the set of coefficients of p(x^{tk}) (respectively, q(x^{tk})). Since p(y)q(y) = 0 and α^{t} = 1_{R}, we have p(x^{tk})q(x^{tk}) = 0 in R[x; α]. Since R is weak α-skew McCoy, by Propositions 2.1 and 2.7, there exists b ≠ 0 such that a_{il} α^{i}(b) ∈ nil(R) for any 0 ⩽ i ⩽ m, 0 ⩽ l ⩽ w_{i}. Thus f_{i}(x)b ∈ nil(R[x; α]). Hence R[x; α] is weak McCoy. □

Let α be an automorphism of a ring R. Suppose that there exists the classical left quotient Q of R. Then for any b^{−1} a ∈ Q, where a, b ∈ R with b regular, the induced map ᾱ : Q(R) → Q(R) defined by ᾱ(b^{−1} a) = (α(b))^{−1} α(a) is also an automorphism.

Proposition 2.8. Assume that there exists the classical left quotient Q of a ring R. If R is reversible, then Q is weak α-skew McCoy if R is weak α-skew McCoy.

Proof. Let f(x) = s_{0}^{−1} a_{0} + s_{1}^{−1} a_{1}x + ⋯+ s_{m}^{−1} a_{m}x^{m} and g(x) = t_{0}^{−1} b_{0} +

t_{1}^{−1} b_{1}x+⋯ +t_{n}^{−1} b_{n}x^{n} ∈ Q[x; ᾱ] such that f(x)g(x) = 0. Let C be a left denominator set. There exist s, t ∈ C and a′_{i}, b′_{j} ∈ R such that s_{i}^{−1} a_{i} = s^{−1} a′_{i} and t_{j}^{−1} b_{j} = t^{−1} b′_{j} for 0 ⩽ i ⩽ m, 0 ⩽ j ⩽ n. Then s^{−1} (a′_{0}+a′_{1}x+⋯ +a′_{m}x^{m})t^{−1} (b′_{0}+b′_{1}x+⋯ +b′_{n}x^{n}) =

0. It follows that (a′_{0}+a′_{1}x+⋯ +a′_{m}x^{m})t^{−1} (b′_{0}+b′_{1}x+⋯ +b′_{n}x^{n}) = 0. Thus (a′_{0}t^{−1} + a′_{1}(α(t))^{−1} x + ⋯+ a′_{m}(α^{m}(t))^{−1} x^{m})(b′_{0} + b′_{1}x + ⋯+ b′_{n}x^{n}) = 0. For (a′_{i}α^{i}(t))^{−1} , there exist t′ ∈ C, a′′_{i}∈ R such that (a′_{i}α^{i}(t))^{−1} = t′a′′_{i}. Hence t^{′−1} (a′′_{0}+a′′_{1}x+⋯ + a′′_{m}x^{m})(b′_{0}+b′_{1}x+⋯ +b′_{n}x^{n}) = 0. We have that (a′′_{0}+a′′_{1}x+⋯ +a′′_{m}x^{m})(b′_{0}+b′_{1}x+⋯ + b′_{n}x^{n}) = 0. Since R is weak α-skew McCoy, there exists b′ ≠ 0 such that a′′_{i}α^{i}(b′) ∈ nil(R). Suppose that (a′′_{i}α^{i}(b′))^{n_{i}} = 0 . Since R is reversible, Q is semicommutative. Then (t^{′−1} (a′′_{i}α^{i}(b′)))^{n_{i}} = 0. So (a′_{i}ᾱ^{i} (t^{−1} b′))^{n_{i}} = ((t^{′−1} a′′_{i})α^{i}(b′))^{n_{i}} = 0. Similarly (s^{−1} a′_{i})(ᾱ^{i} (t^{−1} b′_{j}))^{n_{i}} = 0. Therefore Q is weak α-skew McCoy. □

Acknowledgments. The authors would like to thank the anonymous referee for his/her helpful comments that have improved the presentation of results in this article.

References

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Department of Mathematics (Received 15 07 2012) K. N. Toosi University of Technology (Revised 26 02 2013) P.O. Box 16315 − 1618 Tehran, Iran

nikmehr@kntu.ac.ir

Department of Mathematics Karaj Branch, Islamic Azad university Karaj, Iran

algebra56.tau56@yahoo.com

Department of Mathematics Islamic Azad university, Central Tehran Branch P.O. Box 14168-94351, Tehran, Iran

man.deldar@iauctb.ac.ir