PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 85(99) (2009), 131–137 DOI:10.2298/PIM0999131J

PROPERTIES OF ARMENDARIZ RINGS AND WEAK ARMENDARIZ RINGS

Dušan Jokanović

Communicated by Žarko Mijajlović

Abstract. We consider some properties of Armendariz and rigid rings. We prove that the direct product of rigid (weak rigid), weak Armendariz rings is a rigid (weak rigid), weak Armendariz ring. On the assumption that the factor ring R/I is weak Armendariz, where I is nilpotent ideal, we prove that R is a weak Armendariz ring. We also prove that every ring isomorphism preserves weak skew Armendariz structure. Armendariz rings of Laurent power series are also considered.

1. Introduction

Throughout this paper R denotes an associative ring with identity, σ denotes an endomorphism of R and R[x; σ] denotes a skew polynomial ring with the ordinary addition and the multiplication subject to the relation xr = σ(r)x. When σ is an automorphism, R[x, x^{−1} ; σ] denotes a skew Laurent polynomial ring with the multiplication subject to the relation x^{−1} r = σ^{−1} (r)x.

The notion of Armendariz ring is introduced by Rege and Chhawchharia [1]. They defined a ring R to be Armendariz if f(x)g(x) = 0 implies a_{i}b_{j} = 0, for all polynomials f(x) = ∑ _{i=0}^{n} a_{i}x^{i} and g(x) = ∑ _{j=0}^{m} b_{j} x^{j} from R[x]. The motivation for those rings comes from the fact that Armendariz had shown that reduced rings (a^{2} = 0 implies a = 0) satisfy this condition. The notion of Armendariz ring is natural and useful in understanding the relation between annihilators of rings R and R[x] (see [4]). Those rings were also studied by Armendariz himself, Hong and Kim [5], Chen and Tong [3], Krempa [6] and others.

An endomorphism σ is rigid if aσ(a) = 0 implies a = 0, for all a ∈ R (Krempa [6]). Following Hong, a ring is said to be rigid if it has a rigid endomorphism. Hong also generalized the notions of Armendariz and rigid ring to σ-skew Armendariz ring. Ring R is called σ-skew Armendariz if f(x)g(x) = 0 implies a_{i}σ^{i}(b_{j})= 0, for all f(x) = ∑ _{i=0}^{n} a_{i}x^{i} and g(x) = ∑ _{j=0}^{m} b_{j} x^{j} from R[x; σ] (see [5]). As a generalization of σ-skew Armendariz rings, Ouyang (see [2]) introduced a notion of weak σ-skew Armendariz ring R as a ring in which f(x)g(x) = 0 implies a_{i}σ^{i}(b_{j}) is the nilpotent element of R for all f(x) = ∑ _{i=0}^{n} a_{i}x^{i} and g(x) = ∑ _{j=0}^{m} b_{j} x^{j} from R[x; σ]. Ouyang also introduced a notion of weak σ-rigid ring as a ring with an endomorphism σ that satisfies aσ(a) ∈ nil(R) if and only if a ∈ nil(R) for all a ∈ R where nil(R) is the set of all nilpotent elements of R. In [3] is shown that R is σ-rigid if and only if R is weak σ-rigid and reduced. Here we show that if A is σ_{1}-rigid and B is σ_{2}-rigid, then A × B is γ-rigid, where endomorphism γ is such that γ(a, b) = (σ_{1}(a), σ_{2}(b)). In this paper we consider conditions which characterize σ-rigid rings and prove that R is σ-skew Armendariz ring if and only if R[x, x^{−1} ; σ] is σ-skew Armendariz ring. Chen and Tong (see [3]) have proved that if R and S are rings and σ is an isomorphism of rings R and S and R is α-skew Armendariz ring, then S is σασ^{−1} - skew Armendariz ring. In this paper we prove a variant of this theorem for weak skew Armendariz rings. We also prove that if α is endomorphism of ring R, and the factor ring R[x]/(x^{n}) is weak α̃-skew Armendariz, then V_{n}(R) is weak α̃-skew Armendariz.

2000 Mathematics Subject Classification: Primary 16S36; Secondary 16U90.

2. Rigid rings and weak rigid rings

In this section we give a simple and straightforward proof that the finite direct product of rigid (weak rigid) rings is a rigid (weak rigid) ring. We also show how the notion of rigidity of a ring can be naturally transferred to the notion of rigidity of the corresponding ring of polynomials.

Lemma 2.1. If A is σ_{1}-rigid ring and B is σ_{2}-rigid ring, then A × B is γ-rigid, where γ(a, b) = (σ_{1}(a), σ_{2}(b)).

Proof. Suppose that (a, b)γ(a, b) = (0, 0); then (a, b)(σ_{1}(a), σ_{2}(b)) = (0, 0) so that (aσ_{1}(a), bσ_{2}(b)) = (0, 0). Since aσ_{1}(a) = 0, bσ_{2}(b) = 0, from the fact that A, B are rigid rings we have (a, b) = (0, 0), which means that A × B is a γ-rigid ring. □

Corollary 2.1. Finite direct product of σ_{i}-rigid rings, 1 ⩽ i ⩽ n, is γ-rigid ring, where γ(a_{1}, a_{2}, …, a_{n}) = (σ_{1}(a_{1}), σ_{2}(a_{2}), …, σ_{n}(a_{n})).

Lemma 2.2. If A is a weak σ_{1}-rigid ring and B is a weak σ_{2}-rigid ring, then A × B is a weak γ-rigid ring, where γ is such that γ(a, b) = (σ_{1}(a), σ_{2}(b)).

Proof. Suppose that (a, b)γ(a, b) ∈ nil(A × B). From the definition of γ, we have (a, b)(σ_{1}(a), σ_{2}(b)) ∈ nil(A × B), so that (aσ_{1}(a), bσ_{2}(b)) ∈ nil(A × B) which means that (aσ_{1}(a), bσ_{2}(b))^{n} = (0, 0) for some n ⩾ 2. Therefore (aσ_{1}(a))^{n} = 0, (bσ_{2}(b))^{n} = 0 and aσ_{1}(a) ∈ nil(A), bσ_{2}(b) ∈ nil(B). From the assumption that A is weak σ_{1}-rigid and B weak σ_{2}-rigid we have a ∈ nil(A) and b ∈ nil(B), so that there exist n_{1}, n_{2} such that a^{n_{1}} = 0, b^{n_{2}} = 0. Finally we have (a, b)^{max (n_{1} ,n_{2} )} = (0, 0) which means that (a, b) ∈ nil(A × B).

Conversely, if (a, b) ∈ nil(A × B), using the same arguments we can show that (a, b)γ(a, b) ∈ nil(A × B). □

Corollary 2.2. The finite direct product of weak σ_{i}-rigid rings, 1 ⩽ i ⩽ n, is a weak γ-rigid ring, where γ(a_{1}, a_{2}, …, a_{n}) = (σ_{1}(a_{1}), σ_{2}(a_{2}), …, σ_{n}(a_{n})).

PROPERTIES OF ARMENDARIZ RINGS AND WEAK ARMENDARIZ RINGS 133

We now show how the notion of rigidity naturally transferees from the ring R to the ring R[x]. If σ is an endomorphism of a ring R, then the map σ can be naturally extended to an endomorphism σ′of the ring R[x] by σ′(∑ _{i=0}^{n} a_{i}x^{i}) = ∑ _{i=0}^{n} σ(a_{i})x^{i}.

Theorem 2.1. If R is σ-rigid, then R[x] is σ′-rigid ring.

Proof. Let f(x) = a_{0} + a_{1}x + ⋯+ a_{n}x^{n} and f(x)σ′(f(x)) = 0. We have to prove that f(x) = 0. From the relation

(a_{0} + a_{1}x + ⋯+ a_{n}x^{n})(σ(a_{0}) + σ(a_{1})x + ⋯+ σ(a_{n})x^{n}) = 0,

we have that a_{0}σ(a_{0}) = 0, which means a_{0} = 0. Since the coefficient of x^{2} has to be zero, we have a_{0}σ(a_{2}) + a_{1}σ(a_{1}) + a_{2}σ(a_{0}) = 0, so that a_{1}σ(a_{1}) = 0, and since R is σ-rigid, we have a_{1} = 0. Continuing in this way, since the coefficient of x^{2n−2} has to be zero, and since a_{n−2} = 0, from the previous step, we have

a_{n−2} σ(a_{n}) + a_{n−1} σ(a_{n−1} ) + a_{n}σ(a_{n−2} ) = 0,

which means that a_{n−1} σ(a_{n−1} ) = 0, so that from the rigidity of the ring R we have a_{n−1} = 0. Finally, from the fact that the coefficient of x^{2n} has to be zero, we obtain a_{n}σ(a_{n}) = 0, which means that a_{n} = 0 and so f(x) = 0. □

3. Skew Polynomial Laurent series Rings

In this section we introduce Laurent σ-Armendariz rings and Laurent σ-skew power series rings and we give their useful characterization in terms of σ-skew Armendariz rings. Throughout this section σ is a ring automorphism.

A ring R is a σ-skew Armendariz ring of Laurent type if for every two polynomials f(x) = ∑ _{i=−p}^{q} a_{i}x^{i}, and g(x) = ∑ _{j=−t}^{s} b_{j} x^{j} from R [x, x^{−1} ; σ],

f(x)g(x) = 0 implies a_{i}σ^{i}(b_{j}) = 0, −p ⩽ i ⩽ q, −t ⩽ j ⩽ s.

We say that R is a σ-skew power series Armendariz ring of Laurent type if for every f(x) = ∑ _{i=−p}^{∞} a_{i}x^{i}, and g(x) = ∑ _{j=−t}^{∞} b_{j} x^{j} from the power series ring R[[x, x^{−1} ; σ]],

f(x)g(x) = 0 implies a_{i}σ^{i}(b_{j}) = 0, −p ⩽ i ⩽ ∞, −t ⩽ j ⩽ ∞.

In the following two theorems we give a useful characterization of Laurent σ-skew Armendariz rings and Laurent σ-skew power series rings.

Theorem 3.1. The following conditions are equivalent:

(1) R is a σ-skew Armendariz ring,

(2) R is a σ-skew Armendariz ring of Laurent type.

Proof. Suppose that f(x) = ∑ _{i=−p}^{q} a_{i}x^{i} and g(x) = ∑ _{j=−t}^{s} b_{j} x^{j} are polynomials from the ring R[x, x^{−1} ; σ] such that f(x)g(x) = 0. Since x^{p}f(x) and x^{t}g(x) are polynomials from the ring R[x; σ] we have that x^{p}f(x)g(x)x^{t} = 0 which gives σ^{p}(a_{i})σ^{i+p} (b_{j}) = 0, −p ⩽ i ⩽ q, −t ⩽ j ⩽ s. Since σ is an automorphism,

σ^{p}(a_{i}σ^{i}(b_{j})) = 0,

so that we have a_{i}σ^{i}(b_{j}) = 0. The converse is evident since R[x; σ] ⊂ R [x, x^{−1} ; σ]. □ Theorem 3.2. The following conditions are equivalent:

(1) R is a σ-skew power series Armendariz ring,

(2) R is a σ-skew power series Armendariz ring of Laurent type.

Proof. The same as the proof of the previous theorem. □

We close this section with an interesting remark which gives a sufficient condition for the power series ring R[[x; σ]] to be reduced.

Theorem 3.3. If an endomorphism σ of a reduced ring R satisfies the so-called compatibility condition: aσ(b) = 0 ⇔ ab = 0, then the power series ring R[[x; σ]] is reduced.

Proof. Let f(x) =∑ _{i=0}^{∞} a_{i}x^{i} and (f(x))^{2} = 0. We have to prove that f(x)=0.

It is clear that a_{0}^{2} = 0, so that a_{0} = 0. Now, since the coefficient of x^{2} has to be zero, we have a_{0}a_{2} + a_{1}σ(a_{1}) + a_{2}σ^{2}(a_{0}) = 0, so that we obtain a_{1}σ(a_{1}) = 0. From the compatibility condition we obtain a_{1}^{2} = 0 and since R is reduced, we have a_{1} = 0. Continuing in this way, since the coefficient of x^{2n} is zero, we have a_{n}σ^{n}(a_{n}) = 0 and, using compatibility condition once again, we have a_{n}σ^{n−1} (a_{n}) = 0 and in the same way a_{n}σ(a_{n}) = 0, so that a_{n} = 0. By induction, we have a_{i} = 0, for all i. This means that f(x) = 0 and so the ring R[[x; σ]] is reduced. □

Without compatibility condition the previous theorem is not true. Since if the ring R = Z_{2} ⊕ Z_{2} and σ is defined by σ(a, b) = (b, a), it is easy to check that R[[x; σ]] is not reduced. Observe that (1, 0)(0, 1) = (0, 0) but (1, 0)σ(0, 1) ≠ (0, 0).

4. Weak Armendariz rings

In this section we generalize some results from [3], which are related to σ-skew Armendariz rings, to the weak σ-skew Armendariz case.

A ring R is weak Armendariz if f(x)g(x) = 0 implies a_{i}b_{j} ∈ nil(R) for every two polynomials f(x) = a_{0} + a_{1}x + ⋯+ a_{n}x^{n}, g(x) = b_{0} + b_{1}x + ⋯+ b_{m}x^{m} from the ring R[x]. This definition is equivalent with the fact that ideal 0 is weak Armendariz ideal. We will prove that the class of weak Armendariz rings is closed for direct products. Also, if the factor ring R/I is a weak Armendariz ring, for some nilpotent ideal I, then the ring R is weak Armendariz.

Theorem 4.1. The finite direct product of weak Armendariz rings is a weak Armendariz ring.

Proof. Suppose that R_{1}, R_{2}, …, R_{n} are weak Armendariz rings and R = ∏ _{i=1}^{n} R_{i}. If f(x)g(x) = 0 for some polynomials

f(x) = a_{0} + a_{1}x + a_{2}x^{2} + ⋯+ a_{n}x^{n}, g(x) = b_{0} + b_{1}x + ⋯+ b_{m}x^{m} ∈ R[x],

where a_{i} = (a_{i1}, a_{i2}, …, a_{in}), b_{i} = (b_{i1}, b_{i2}, …, b_{in}) are elements of the product ring R, define

f_{k}(x) = a_{0k} + a_{1k}x + ⋯+ a_{nk} x^{n}, g_{k}(x) = b_{0k} + b_{1k}x + ⋯+ b_{mk} x^{m}.

From f(x)g(x) = 0, we have a_{0}b_{0} = 0, a_{0}b_{1} + a_{1}b_{0} = 0, …, a_{n}b_{m} = 0, and this implies

a_{01}b_{01} = a_{02}b_{02} = ⋯= a_{0n} b_{0n} = 0
a_{01}b_{11} + a_{11}b_{01} = ⋯= a_{0n} b_{1n} + a_{1n} b_{0n} = 0
a_{n1} b_{m1} = a_{n2} b_{m2} = ⋯= a_{nn} b_{mn} = 0.

This means that f_{k}(x)g_{k}(x) = 0 in R_{k}[x], 1 ⩽ k ⩽ n, and since R_{k} are weak Armendariz rings, we have a_{ik} b_{jk} ∈ nil(R_{k}). Now, for each i, j, there exists positive integers m_{ijk} such that (a_{ik} b_{jk})^{m_{ijk}} = 0 in the ring R_{k}, 1 ⩽ k ⩽ n. If we take m_{ij} = max{m_{ijk} : 1 ⩽ k ⩽ n}, then it is clear that (a_{i}b_{j})^{m_{ij}} = 0 and this means that R is a weak Armendariz ring. □

Theorem 4.2. If I is a nilpotent ideal of ring R such that R/I is a weak Armendariz ring, then R is a weak Armendariz ring.

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

are polynomials from R[x] such that f(x)g(x) = 0. This implies

(\overline{a_{0}} + \overline{a_{1}}x + ⋯+ \overline{a_{n}}x^{n})(\overline{b_{0}}+ \overline{b_{1}}x + ⋯+ \overline{b_{m}}x^{m}) = 0,

and since R/I is weak Armendariz, we have that \overline{a_{i}} \overline{b_{j}} ∈ nil(R|I). From the fact that the ideal I is nilpotent, we obtain that a_{i}b_{j} ∈ nil(R). □

Recall that a ring R is weak σ-rigid if aσ(a) ∈ nil(R) ⇔ a ∈ nil(R). It is easy to see that the notion of weak σ-rigid ring generalizes the notion of a σ-rigid ring. Every homomorphism σ of rings R and S can be extended to the homomorphism of rings R[x] and S[x] by ∑ _{i=0}^{m} a_{i}x^{i} ↦ ∑ _{i=0}^{m} σ(a_{i})x^{i}, which we also denote by σ. Chen and Tong in [3] prove that if σ is a ring isomorphism of rings R and S and R is α-skew Armendariz, then S is a σασ^{−1} skew Armendariz ring. We prove the weak skew Armendariz variant of this theorem.

Theorem 4.3. Let R and S be rings with a ring isomorphism σ : R → S. If R is weak α-skew Armendariz, then S is weak σασ^{−1} -skew Armendariz.

Proof. Let f(x) = ∑ _{i=0}^{m} a_{i}x^{i} and g(x) = ∑ _{j=0}^{m} b_{j} x^{j} are polynomials from the ring S[x; σασ^{−1} ]. We have to prove that f(x)g(x) = 0 implies a_{i}(σασ^{−1} )^{i}(b_{j}) ∈ nil(S), for all i and j.

As we noted, σ extends to the isomorphism of the corresponding polynomial rings, so that there exist polynomials f_{1}(x) = ∑ _{i=0}^{m} a′_{i}x^{i} and g_{1}(x) = ∑ _{j=0}^{m} b′_{j}x^{j} from R[x] 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 shall show that f(x)g(x) = 0 implies f_{1}(x)g_{1}(x) = 0. If f(x)g(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) = 0 in the ring R[x; α].

It remains to prove that f_{1}(x)g_{1}(x) = 0 implies a_{i}(σασ^{−1} )^{i}(b_{j}) ∈ nil(S). From the fact that R is weak α-skew Armendariz we have a′_{i}α^{i}(b′_{j}) ∈ nil(R), and since a′_{i} = σ^{−1} (a_{i}), b′_{j} = σ^{−1} (b_{j}), we have σ^{−1} (a_{i})α^{i}σ^{−1} (b_{j}) ∈ nil(R). This implies

σ^{−1} (a_{i})σ^{−1} σα^{i}σ^{−1} (b_{j}) = σ^{−1} (a_{i}(σασ^{−1} )^{i}(b_{j})) ∈ nil(R)

and finally we obtain

a_{i}(σασ^{−1} )^{i}(b_{j}) ∈ nil(S), 0 ⩽ i, j ⩽ m.

Hence S is weak σασ^{−1} -skew Armendariz. □

In our closing result, we shall show that, under certain condition, the subring of upper triangular skew matrices over a ring R has a weak skew Armendariz structure.

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 4.1. Suppose that α is an endomorphism of ring R. If the factor ring R[x]/(x^{n}) is weak α̃-skew Armendariz, then V_{n}(R) is weak α̃-skew Armendariz.

Proof. Suppose that R[x]/(x^{n}) is weak α̃-skew Armendariz and define the ring isomorphism θ : V_{n}(R) → R[x]/(x^{n}) 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} α̃θ-skew Armendariz and

θ^{−1} α̃θ(r_{0}I_{n} + r_{1}V + ⋯+ r_{n−1} V^{n−1} )
= θ^{−1} α̃(r_{0} + r_{1}x + ⋯+ r_{n−1} x^{n−1} + (x^{n}))
= θ^{−1} (α(r_{0}) + α(r_{1})x + ⋯+ α(r_{n−1} )x^{n−1} + (x^{n}))
= α(r_{0})I_{n} + α(r_{1})V + ⋯+ α(r_{n−1} )V^{n−1}
= α̃(r_{0}I_{n} + r_{1}V + ⋯+ r_{n−1} V^{n−1} ),

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

References

[1] M. R. Rege, S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A. Math. Sci. 73 (1997), 14–17

[2] L. Ouyang, Extensions of generalized α-rigid rings, Internat. J. Algebra 3 (2008), 105–116

[3] W. Chen, W. Tong, On skew Armendariz and rigid rings, Houston J. Math. 22(2) (2007)

[4] Y. Hirano, On annihilator ideals of polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 151(3) (2000), 105–122 [5] C. Y. Hong, N. K. Kim, T. K. Kwak, On skew Armendariz rings, Comm. Algebra 31(2) (2003),

105–122

[6] J. Krempa, Some examples of reduced rings, Algebra Colloq. 3(4) (1996), 289–330

Prirodno-matematički fakultet (Received 17 08 2008) 81000 Podgorica (Revised 06 02 2009) Montenegro

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