Volume 2007, Article ID 61549,7pages doi:10.1155/2007/61549
Research Article
On Classical Quotient Rings of Skew Armendariz Rings
A. R. Nasr-Isfahani and A. Moussavi
Received 7 February 2007; Accepted 2 August 2007
Recommended by Howard E. Bell
LetRbe a ring,αan automorphism, andδanα-derivation ofR. If the classical quotient ringQof Rexists, thenRis weakα-skew Armendariz if and only ifQis weak α-skew Armendariz.
Copyright © 2007 A. R. Nasr-Isfahani and A. Moussavi. This is an open access article dis-tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop-erly cited.
1. Introduction
For a ringRwith a ring endomorphism,α:R→Rand anα-derivationδofR, that is,δis an additive map such thatδ(ab)=δ(a)b+α(a)δ(b), for alla,b∈R, we denote byR[x;α,δ] the skew polynomial ring whose elements are the polynomials overR, the addition is defined as usual and the multiplication subject to the relationxa=α(a)x+δ(a) for any
a∈R.
Rege and Chhawchharia [1] called a ringRan Armendariz ring if whenever any poly-nomial f(x)=a0+a1x+···+anxn,g(x)=b0+b1x+···+bmxm∈R[x] satisfy f(x)g(x)=
0, thenaibj=0 for eachi,j. This nomenclature was used by them since it was
Armen-dariz [2, Lemma 1] who initially showed that a reduced ring (i.e., a ring without nonzero nilpotent elements) always satisfies this condition. A number of papers have been written on the Armendariz property of rings. For basic and other results on Armendariz rings, see, for example, [1–11].
The Armendariz property of rings was extended to skew polynomial rings with skewed scalar multiplication in [7].
For an endomorphismαof a ringR,Ris called anα-skew Armendariz ring (or, a skew
Armendariz ring with the endomorphismα) if forp=m
i=0aixi andq=nj=0 bjxj∈
Recall that an endomorphismαof a ringRis called rigid (see [5] and [12]) ifaα(a)=0 impliesa=0 fora∈R.Ris called anα-rigid ring [12] if there exists a rigid endomor-phism αof R. Note that any rigid endomorphism of a ring is a monomorphism, and
α-rigid rings are reduced by [12, Propositions 5 and 6]. IfRis anα-rigid ring, then forp=m
i=0aixiandq=nj=0bjxj∈R[x;α,δ],pq=0 if
and only ifaibj=0 for all 0≤i≤mand 0≤j≤n[12, Proposition 6].
Various properties of the Ore extensions have been investigated by many authors; (see [1–13]). Most of these have worked either with the caseδ=0 andα-automorphism or the case whereαis the identity. However, the recent surge of interest in quantum groups and quantized algebras has brought renewed interest in general skew polynomial rings, due to the fact that many of these quantized algebras and their representations can be expressed in terms of iterated skew polynomial rings. This development calls for a thorough study of skew polynomial rings.
Anderson and Camillo [3] assert that for a semiprime left and right Noetherian ring
R,Ris Armendariz if and only if the classical right quotient ringQ(R) ofRis reduced. Anderson and Camillo [3, Theorem 7] proved that ifRis a prime ring which is left and right Noetherian, thenRis Armendariz if and only ifRis reduced. Kim and Lee in [9] obtained this result under a weaker condition. They proved that ifRis a semiprime right and left Goldie ring, thenRis Armendariz if and only ifRis reduced. Kim and Lee also proved that if there exists the classical right quotient ringQ(R) of a ringR, thenR is reduced if and only ifQ(R) is reduced.
In this paper, we obtain a generalized result of [3, Theorem 7] and [9, Theorem 16], for reduced rings, ontoα-skew Armendariz rings [9, Proposition 18 and Corollary 19] as corollaries.
2. The results
We first give the following definition ofα-skew Armendariz ring, and notice that our definition is compatible with Hong et al.’s [7], assumingδto be the zero mapping.
Definition 2.1. LetRbe a ring with a ring endomorphismαand anα-derivationδ.Ris anα-skew Armendariz ring (or, a skew Armendariz ring with the endomorphismα) if for
p=m
i=0aixiandq=nj=0bjxj∈R[x;α,δ],pq=0 impliesaiαi(bj)=0 for all 0≤i≤m
and 0≤j≤n.
We notice that to extend Armendariz property to the Ore extensionR[x;α,δ], we do not need any more conditions. In the caseδ=0, several examples ofα-skew Armendariz rings are obtained in [7], and in the same method, one can provide similar results as in [7], for the more general cases, of the Ore extensionR[x;α,δ]. For instance, it is easy to prove thatα-rigid rings areα-skew Armendariz.
Definition 2.2. LetR be a ring with a ring endomorphismαand anα-derivationδ.R
is a weakα-skew Armendariz ring, if for linear polynomials f(x)=a0+a1x andg(x)= b0+b1x∈R[x;α,δ], f(x)g(x)=0 impliesaiαi(bj)=0 for all 0≤i, j≤1.
endomorphismIRof a ringR,Ris Armendariz if and only ifRisIR-skew Armendariz and
δ=0.
LetRbe a ring with the classical right (left) quotient ringQ. Then each injective en-domorphismαandα-derivationδofRextends toQ, respectively, by settingα(c−1r)= α(c)−1α(r) andδ(c−1r)=α(c)−1
(δ(r)−δ(c)c−1r), for eachr,c∈Rwithcregular.
A ringRis called right Ore givena,b∈Rwithbregular if there exista1,b1∈Rwithb1
regular such thatab1=ba1. It is a well-known fact thatRis a right Ore ring if and only if
there exists the classical right quotient ring ofR.
Note that every Ore domain is anα-skew Armendariz ring for every automorphismα
andα-derivationδ.
Now we obtain a generalized result of [9, Theorem 16], for reduced rings, ontoα-skew Armendariz rings, by showing that the weakα-skew Armendariz condition extends to its classical quotient ring.
Theorem 2.3. LetR be a ring, αan automorphism, and δ an α-derivation ofR. If the classical quotient ringQofRexists, thenRis weakα-skew Armendariz if and only ifQis weakα-skew Armendariz.
Proof. Let f(x)=c−1
0 a0+c−11a1xandg(x)=s−01b0+s1−1b1x∈Q[x;α,δ] such that f(x)g(x)=
0. Then there existai,bj∈R and regular elementsc,s∈Rsuch thatc−i1ai=c−1ai and
s−1
j bj=s−1bj. We then have (c−1a0+c−1a1x)(s−1b0+s−1b1x)=0. So (a0+a1x)s−1(b0+ b1x)= 0 and hence [a0s−1 +a1δ(s−1) +a1α(s−1)x](b0+ b1x)= 0. Then [a0s−1− a1α(s)−1δ(s)s−1+a1α(s)−1x](b0+b1x)=0. Now there existd,s2∈R, withs2regular, such
thatδ(s)s−1=s−1
2 d. Thus [a0s−1−a1α(s)−1s−21d+a1α(s)−1x](b0+b1x)=0. There exist a0,a1,a1 ,s3,s4,s5∈R, with s3, s4, s5 regulars, such that a0s−1=s−31a0,a1α(s)−1s−21= s−1
4 a1,a1α(s)−1=s5−1a1 . So [s−31a0 −s4−1a1d+s−51a1x](b0+b1x)=0. There existt,d0,d1, d2∈R, with t regular, such that s−31a0 =t−1d0,s4−1a1 =t−1d1,s5−1a1 =t−1d2. Hence t−1(d
0−d1d+d2x)(b0+b1x)=0. Now the Armendariz condition implies the following
equations:
d0−d1d
b0=0,
d0−d1d
b1=0,
d2α
b0
=0,
d2α
b1
=0.
(2.1)
We have (d0−d1d+d2x)(b0+b1x)=0, so (d0−d1d)b0+d2δ(b0) + ((d0−d1d)b1+ d2α(b0) +d2δ(b1))x+d2α(b1)x2=0. Thus we get the following equations:
d0−d1d
b0+d2δ
b0
=0,
d0−d1d
b1+d2α
b0
+d2δ
b1
=0,
d2α
b1
=0.
These equations and (2.1) imply that
d2δ
b0
=0,
d2δ
b1
=0. (2.3)
Now we have
d0−d1d
b0=0⇐⇒t−1
d0−d1d
b0=0⇐⇒
s−1
3 a0 −s−41a1d
b0=0 ⇐⇒a0s−1−a1α(s)−1s−21d
b0=0⇐⇒
a0s−1−a1α(s)−1δ(s)s−1
b0=0
⇐⇒a0s−1+a1 δ
s−1b
0=0.
(2.4)
A similar argument shows that
d0−d1d
b1=0⇐⇒
a0s−1+a1δ
s−1b
1=0. (2.5)
Also we have
d2δ
b0
=0⇐⇒t−1d 2δ
b0
=0⇐⇒s−1 5 a1 δ
b0
=0⇐⇒a1α(s)− 1
δb0
=0
⇐⇒a1 α
s−1 δb
0
=0. (2.6)
A similar argument shows that
d2δ
b1
=0⇐⇒a1 α
s−1 δb1
=0, (2.7)
d2αb0
=0⇐⇒a1 α
s−1αb0
=0, (2.8)
d2αb1
=0⇐⇒a1 α
s−1αb
1
=0. (2.9)
Now take h(x)=a0+a1x and k(x)=s−1b1∈Q[x;α,δ], and using (2.9), (2.5),
(2.7), then we get
h(x)k(x)=a0+a1x
s−1b
1
=a0s−1b1+a1α
s−1b
1
x+a1δ
s−1b
1
=a0s−1b1+a1δ
s−1b
1
+a1α
s−1αb
1
x
=a0s−1b1+a1δ
s−1b1+a1α
s−1 δb1
=a0s−1+a1δ
s−1b
1+a1α
s−1 δb
1
=0.
(2.10)
Therefore, we have (a0+a1x)(s−1b1)=0. Now there existm,n∈Rwithnregular, we get s−1b
1=mn−1. Thus (a0+a1x)mn−1=0, hence (a0+a1x)m=0. SinceRis weak
skew-Armendariz, we can deduce thata0m=a1δ(m)=0. But
Equations (2.5) and (2.11) imply that
a1δ
s−1b
1=0. (2.12)
Now takep(x)=a0+a1xandq(x)=s−1b0, using (2.8), (2.4), (2.6) then we get
p(x)q(x)=a0+a1x
s−1b
0=a0s−1b0+a1α
s−1b
0
x+a1δ
s−1b
0
=a0s−1b0+a1δ
s−1b
0+a1α
s−1 δb
0
+a1α
s−1αb
0
x
=a0s−1+a1δ
s−1b
0+a1α
s−1 δb
0
=0.
(2.13)
So (a0+a1x)s−1b0=0. But there exist u,v∈Rwithv regular such thats−1b0=uv−1.
Thus (a0+a1x)uv−1=0 and hence (a0+a1x)u=0. The Armendariz condition implies
thata0u=0, and soa0uv−1=0. So we get
a0s−1b0=0. (2.14)
By (2.14) and (2.4), we have
a1δ
s−1b
0=0. (2.15)
By (2.14),
a0s−1b0=0⇐⇒c−1a0s0−1b0=0⇐⇒c−01a0s−01b0=0. (2.16)
By (2.11),
a0s−1b1=0⇐⇒c−1a0s1−1b1=0⇐⇒c−01a0s−11b1=0. (2.17)
By (2.6) and (2.15), we havea1δ(s−1)b0=0=a1α(s−1)δ(b0). Thus
a1 δ
s−1b0+α
s−1)δb0
=0⇐⇒a1δ
s−1b0
=0⇐⇒c−1a1δ
s−1b0
=0
⇐⇒c−11a1δ
s−01b0
=0.
(2.18)
By (2.12) and (2.7), we havea1δ(s−1)b1=0=a1α(s−1)δ(b1). Thus
a1δ
s−1b
1+a1α
s−1δb
1
=0⇐⇒a1δ
s−1b
1
=0⇐⇒c−1 1 a1δ
s−1
1 b1
=0. (2.19)
Using (c−01a0+c1−1a1x)(s−01b0+s−11b1x)=0 and (2.16), (2.17), (2.18), (2.19), we also
havec−11a1α(s0−1b0)=0 andc1−1a1α(s−11b1)=0. ThereforeQis a weakα-skew Armendariz
ring.
Now we show that skew-Armendariz rings are Abelian (i.e., every idempotent is cen-tral).
Proof. LetRbe a weakα-skew Armendariz ring and lete2=e,a∈R. Consider the
poly-nomials f(x)=e−ea(1−e)xandg(x)=1−e+ea(1−e)x∈R[x;α,δ]. Then we have
f(x)g(x)=0.Since Ris weak skew Armendariz,eea(1−e)=0.So ea=eae.Next let
h(x)=1−e−(1−e)aex andk(x)=e+ (1−e)aex∈R[x;α,δ]. We have h(x)k(x)=0 and sinceRis weak skew Armendariz, it implies that (1−e)(1−e)ae=0. Thusae=eae
and soae=eawhich implies thatRis Abelian.
Corollary 2.5. Let R be a semiprime Goldie ring and α-automorphism and δ an α -derivation ofR. Then the following are equivalent:
(1)Ris weakα-skew Armendariz;
(2)Risα-skew Armendariz;
(3)Qisα-skew Armendariz;
(4)Qis weakα-skew Armendariz;
(5)Risα-rigid;
(6)Qisα-rigid.
Proof. The proof follows byTheorem 2.3. For the implication 2⇒5, notice that whenR
is a weakα-skew Armendariz ring, then byTheorem 2.3,Qis weakα-skew Armendariz and henceQis Abelian byLemma 2.4soQis an aAbelian semisimple ring and hence is reduced. Now, suppose thataα(a)=0 fora∈Q. So we haveδ(a)α(a)=α(a)δ(α(a))=
0. Now, let h(x)=α(a)−α(a)x and k(x)=a+α(a)x∈Q[x;α,δ]. Thenh(x)k(x)=0. SinceQis weakα-skew Armendariz, we haveα(a)α(a)=0. ButQis reduced andαis a
monomorphism, thereforea=0. ThusQisα-rigid, soRisα-rigid.
ByCorollary 2.5, it is shown that a semiprime right Goldie ringRwith an automor-phismαis weakα-skew Armendariz if and only if it is reduced.
Acknowledgment
The authors are thankful to the referee for a careful reading of the paper and for some helpful comments and suggestions.
References
[1] M. B. Rege and S. Chhawchharia, “Armendariz rings,” Proceedings of the Japan Academy, Series A, vol. 73, no. 1, pp. 14–17, 1997.
[2] E. P. Armendariz, “A note on extensions of Baer and p.p.-rings,” Journal of the Australian Math-ematical Society, Series A, vol. 18, pp. 470–473, 1974.
[3] D. D. Anderson and V. Camillo, “Armendariz rings and Gaussian rings,” Communications in Algebra, vol. 26, no. 7, pp. 2265–2272, 1998.
[4] E. Hashemi and A. Moussavi, “Polynomial extensions of quasi-Baer rings,” Acta Mathematica Hungarica, vol. 107, no. 3, pp. 207–224, 2005.
[5] Y. Hirano, “On annihilator ideals of a polynomial ring over a noncommutative ring,” Journal of Pure and Applied Algebra, vol. 168, no. 1, pp. 45–52, 2002.
[6] C. Y. Hong, T. K. Kwak, and S. T. Rizvi, “Extensions of generalized Armendariz rings,” Algebra Colloquium, vol. 13, no. 2, pp. 253–266, 2006.
[8] C. Huh, Y. Lee, and A. Smoktunowicz, “Armendariz rings and semicommutative rings,” Com-munications in Algebra, vol. 30, no. 2, pp. 751–761, 2002.
[9] N. K. Kim and Y. Lee, “Armendariz rings and reduced rings,” Journal of Algebra, vol. 223, no. 2, pp. 477–488, 2000.
[10] T.-K. Lee and T.-L. Wong, “On Armendariz rings,” Houston Journal of Mathematics, vol. 29, no. 3, pp. 583–593, 2003.
[11] A. Moussavi and E. Hashemi, “On (α,δ)-skew Armendariz rings,” Journal of the Korean Mathe-matical Society, vol. 42, no. 2, pp. 353–363, 2005.
[12] C. Y. Hong, N. K. Kim, and T. K. Kwak, “Ore extensions of Baer and p.p.-rings,” Journal of Pure and Applied Algebra, vol. 151, no. 3, pp. 215–226, 2000.
[13] A. Moussavi and E. Hashemi, “Semiprime skew polynomial rings,” Scientiae Mathematicae Japonicae, vol. 64, no. 1, pp. 91–95, 2006.
A. R. Nasr-Isfahani: Department of Mathematics, Tarbiat Modares University, P.O. Box 14115-170, Tehran, Iran
Email address:a nasr isfahani@yahoo.com
A. Moussavi: Department of Mathematics, Tarbiat Modares University, P.O. Box 14115-170, Tehran, Iran