Vol. 7, No. 4, 2014, 387-394
ISSN 1307-5543 – www.ejpam.com
Skew-Laurent rings over
σ
(
∗
)
-rings
V. K. Bhat
School of Mathematics, SMVD University, P/o SMVD University, Katra, J and K, India- 182320
Abstract. LetRbe an associative ring with identity 16=0, andσan endomorphism ofR. We recall
σ(∗)property onR(i.e. aσ(a)∈P(R)impliesa∈P(R)fora∈R, where P(R)is the prime radical of
R). Also recall that a ringRis said to be 2-primal if and only ifP(R)and the set of nilpotent elements ofRcoincide, if and only if the prime radical is a completely semiprime ideal. It can be seen that a
σ(∗)-ring is a 2-primal ring.
LetRbe a ring andσan automorphism ofR. Then we know thatσcan be extended to an automorphism (sayσ) of the skew-Laurent ringR[x,x−1;σ]. In this paper we show that ifRis a Noetherian ring and
σis an automorphism ofRsuch thatRis aσ(∗)-ring, thenR[x,x−1;σ]is aσ(∗)-ring. We also prove a similar result for the general Ore extensionR[x;σ,δ], whereσis an automorphism ofRandδ a
σ-derivation ofR.
2010 Mathematics Subject Classifications: 16-XX; 16N40, 16P40, 16S36.
Key Words and Phrases: Minimal prime, prime radical, automorphism,σ(∗)-ring
1. Introduction
A ring Ralways means an associative ring with identity 16=0. The set of prime ideals of R is denoted byS pec(R). The sets of minimal prime ideals ofRis denoted by M in.S pec(R). Prime radical and the set of nilpotent elements ofRare denoted byP(R)andN(R)respectively. LetR be a ring andσan automorphism ofR. Let I be an ideal ofRsuch that σm(I) = I for somem∈N (whereNis the set of positive integers). We denote∩mi=1σi(I)by I0. The field of rational numbers is denoted by Q and the field of real numbers is denoted by R unless otherwise stated.
This article concerns the study of skew-Laurent rings overσ(∗)-rings, whereσis an auto-morphism ofR.
Email address:[email protected]
σ
(
∗
)
-rings
Recall that in Krempa[8], a ringRis calledσ-rigid if there exists an endomorphismσof Rwith the property thataσ(a) =0 impliesa=0 fora∈R. In[9], Kwak defines aσ(∗)-ring Rto be a ring in whichaσ(a)∈P(R)impliesa∈P(R)fora∈R.
Example 1. Let R=
F F
0 F
, where F is a field. Then P(R) =
0 F 0 0
. Letσ:R→R be
defined byσ
a b 0 c = a 0 0 c
. Then it can be seen thatσis an endomorphism of R
and R is aσ(∗)-ring.
2-primal Rings
We do not want to talk about 2-primal rings, but because of a close relation between a
σ(∗)-ring and a 2-primal ring, we have the following:
Recall that a ringR is 2-primal if and only if N(R) = P(R), i.e. if the prime radical is a completely semiprime ideal. An ideal I of a ring Ris called completely semiprime if a2 ∈I impliesa∈I fora∈R. We note that a commutative ring is 2-primal and so is a reduced ring. 2-primal rings have been studied in recent years and the 2-primal property is being studied for various types of rings. In[10], Greg Marks discusses the 2-primal property ofR[x;σ,δ], whereRis a local ring,σis an automorphism of R andδis aσ-derivation ofR. He has proved that whenRis a local ring with a nilpotent maximal ideal, the Ore extensionR[x;σ,δ]will or will not be 2-primal depending on theδ-stability of the maximal ideal ofR.
In[9], Kwak establishes a relation between a 2-primal ring and aσ(∗)-ring. It has been proved that ifRis a ring andσan endomorphism ofRsuch thatσ(P(R))⊆ P(R), thenRis a
σ(∗)-ring implies thatRis 2-primal. Therefore, we see that ifRis a Noetherian ring andσan automorphism ofR, thenRis aσ(∗)-ring implies thatRis 2-primal.
The following example shows that if Ris a Noetherian ring, then evenR[x] need not be 2-primal.
Example 2. Let R= M2(Q), the set of2×2matrices overQ. Then R[x] is a prime ring with non-zero nilpotent elements and, so can not be 2-primal.
Skew Polynomial Rings
LetRbe a ring,σbe an endomorphism ofRandδaσ-derivation ofR. Recall thatδis an additive mapδ:R→Rsuch thatδ(a b) =δ(a)σ(b) +aδ(b), for alla,b∈R.
Example 3. Letσbe an automorphism of a ring R andδ:R→R any map. Letφ:R→M2(R) defined byφ(r) =
σ(r) 0
δ(r) r
, for all r∈R be a homomorphism. Thenδis aσ-derivation of
Recall that the skew polynomial ring (Ore extension)R[x;σ,δ]is the usual ring of polyno-mials with coefficients inR, in which multiplication is subject to the relationa x =xσ(a)+δ(a) for alla ∈R. We take any f(x)∈R[x;σ,δ]to be of the form f(x) =Pni=0xiai. We denote R[x;σ,δ]by O(R). If I is an ideal ofRsuch thatσ(I) = I andδ(I)⊆ I, thenO(I) denotes I[x;σ,δ], which is an ideal ofO(R).
Skew-Laurent Rings
Recall that R[x,x−1;σ] is the usual ring of Laurent polynomials with coefficients in R, in which multiplication is subject to the relation a x = xσ(a) for all a ∈ R. We take any f(x)∈R[x,x−1;σ]to be of the form f(x) =Pin=−mxiai. We denoteR[x,x−1;σ]by L(R). If
an idealIof a ringRisσ-stable (i.e. σ(I) =I), then we denote as usual I[x,x−1;σ]byL(I). We also note that ifσis an automorphism ofR, then it can be extended to an automorphism (sayσ) ofR[x,x−1;σ]such thatσ(x) = x; i.e. σ(Σni=−mxiai) = Σni=−mx
iσ(a
i). The study
of skew polynomial rings and skew-Laurent rings has been of interest to many authors. For example[1, 6, 7, 9].
In this paper we prove the following results:
Theorem 2: LetRbe a Noetherian ring andσan automorphism ofR. ThenRis aσ(∗)-ring if and only ifR[x,x−1;σ]is aσ(∗)-ring.
Theorem 3: LetRbe a Noetherian ring which is also an algebra over Q. Letσbe an automor-phism ofRsuch thatRis aσ(∗)-ring andδaσ-derivation of R such that
σ(δ(a)) =δ(σ(a))for alla∈R. ThenR[x;σ,δ]is aσ(∗)-ring.
2. Preliminaries
We begin this section with the following Proposition:
Proposition 1. Let R be a ring andσan automorphism of R. Then R is aσ(∗)-ring implies R is 2-primal.
Proof. Leta∈Rbe such thata2∈P(R). Then
aσ(a)σ(aσ(a)) =aσ(a)σ(a)σ2(a)∈σ(P(R)) =P(R).
Thereforeaσ(a)∈P(R)and hencea∈P(R).
The following example shows that there exists an endomorphismσof a ringR such that the converse of the above Proposition does not hold.
Example 4. Let R=F[x], F a field. Then R is a commutative domain, and therefore is 2-primal with P(R) =0. Letσ:R→R be defined byσ(f(x)) = f(0). Let f(x) =x a,06=a∈F . Then
Before we give a characterization of a Noetherianσ(∗)-ring, we require the following: Recall that an idealPof a ringRis completely prime ifR/Pis a domain, i.e. a b∈Pimplies a∈Por b∈Pfora, b∈R(McCoy[11]).
Note that a completely prime ideal is a prime ideal, but the converse need not be true.
For example, letR=
Z Z Z Z
=M2(Z). If pis a prime number, then the ideal
P = M2(pZ) is a prime ideal of R, but is not strongly prime, since for a =
1 0 0 0
and
b=
0 0 0 1
we havea b∈P, even thougha∈/P andb∈/P.
Proposition 2(Proposition 2.1 of Bhat[6]). Let R be a Noetherian ring, andσan automorphism of R. Then R is aσ(∗)-ring if and only if for each minimal prime U of R,σ(U) =U and U is a completely prime ideal of R.
Proof. To make the article self contained, we give a proof (a modified one):
LetR be a Noetherian ring such that for each minimal primeU ofR,σ(U) =U andU is completely prime ideal ofR. Leta∈Rbe such thataσ(a)∈P(R) =∩n
i=1Ui, whereUi are the
minimal primes of R. Now for each i, a∈Ui or σ(a)∈Ui as Ui are completely prime. Now
σ(a)∈Ui =σ(Ui)implies thata∈Ui. Thereforea∈P(R). HenceRis aσ(∗)-ring.
Conversely, suppose thatRis aσ(∗)-ring and letU=U1be a minimal prime ideal ofR. Now by Proposition 1,P(R)is completely semiprime. NowM in.S pec(R)is finite by Theorem (2.4) of Goodearl and Warfield[7]. Let U2,U3, . . . ,Un be the other minimal primes ofR. Suppose
thatσ(U)6=U. Thenσ(U)is also a minimal prime ideal ofR. Renumber so thatσ(U) =Un. Let a∈ ∩ni=−11Ui. Thenσ(a)∈Un, and soaσ(a)∈ ∩ni=1Ui = P(R). Therefore a∈ P(R), and thus ∩ni=−11Ui ⊆ Un, which implies that Ui ⊆ Un for some i 6= n, which is impossible. Hence σ(U) =U.
Now since aσ(∗)-ring is 2-primal, minimal prime ideals are completely prime. HenceU is completely prime.
Note that in above Theorem the condition of completely primeness of minimal prime ideals can not be deleted. Towards this we have the following:
Remark 1. Let R be a Noetherian ring andσ an automorphism of R such thatσ(U) = U for each minimal prime ideal U of R. Then R need not be aσ(∗)-ring (Example 4).
3. Skew-Laurent Rings Over
σ
(
∗
)
-rings
Goodearl and Warfield proved in (2ZA) of[7]that ifRis a commutative Noetherian ring, and ifσis an automorphism of R, then an ideal I ofRis of the form P∩R for some prime idealPofR[x,x−1;σ]if and only if there is a prime idealSofRand a positive integermwith
σm(S) =S, such thatI =∩σi(S),i=1, 2, . . . ,m.
all j∈N. Therefore, there exists somem∈Nsuch thatσm(U) =U for allU ∈M in.S pec(R). We denote∩mi=1σi(U)byU0.
We now have the following:
Theorem 1. Let R be a Noetherian ring andσan automorphism of R. Then P ∈M in.S pec(L(R)) if and only if there exists U ∈M in.S pec(R) such that L(P∩R) = (P∩R)[x,x−1;σ] = P and P∩R=U0.
Proof. See Theorem (2.4) of Bhat[1].
As mentioned in the introduction, we note that ifσis an automorphism ofR, then it can be extended to an automorphism (sayσ) ofR[x,x−1;σ]such thatσ(x) =x; i.e.
σ(Σn
i=−mxiai) = Σin=−mxiσ(ai).
With this we are now in a position to prove the following Theorem:
Theorem 2. Let R be a Noetherian ring andσan automorphism of R. Then R is aσ(∗)-ring if and only if L(R) =R[x,x−1;σ]is a Noetherianσ(∗)-ring.
Proof. LetRbe a Noetherian ring,σan automorphism ofRsuch thatRis aσ(∗)-ring and
δaσ-derivation of R. We shall prove that O(R) =R[x;σ,δ]is a Noetherian σ(∗)-ring. For this we will show that any minimalP∈M in.S pec(O(R))is completely prime andσ(P) =P.
Let P ∈ M in.S pec(O(R)). Then by Theorem 1, there exists U ∈ M in.S pec(R) such that P=U0[x,x−1;σ]. NowRis aσ(∗)-ring implies thatσ(U) =Uby Proposition 2, and therefore U0=U. So P=U[x,x−1;σ]and thusσ(P) =P.
We now show that P= U[x,x−1;σ] is completely prime. Nowσcan be extended to an automorphism ofR/U in a natural way. We note thatO(R)/P∼= (R/U)[x,x−1;σ], and since U is completely prime, R/U is a domain and so(R/U)[x,x−1;σ] is also a domain. Hence P=U[x,x−1;σ]is completely prime.
Thusσ(P) =P andPis completely prime for allP∈M in.S pec(L(R)). Moreover
L(R) =R[x,x−1;σ]is Noetherian by Theorem (1.17) of Goodearl and Warfield[7]. Hence by Proposition 2R[x,x−1;σ]is aσ(∗)-ring.
Conversely letL(R) =R[x,x−1;σ]be aσ(∗)-ring. LetU∈M in.S pec(R). Then Theorem 1 implies thatL(U0)∈M in.S pec(L(R)). NowL(R)be aσ(∗)-ring implies that
σ(L(U0)) = L(U0)andL(U0)is completely prime ideal of L(R). Now there is an embedding R/(L(U0)∩R)→ L(R)/L(U0). Since L(R)/L(U0) is an integral domain, so isR/(L(U0)∩R). Therefore, U0 = L(U0)∩R) is a completely prime ideal of R. Now U0 ⊆ U implies that U0 =U. Soσ(U) =U andU is a completely prime ideal ofR. Hence by Proposition 2Ris a
σ(∗)-ring.
Remark 2.
ii) If R is 2-primal Noetherian ring, then R[x,x−1;σ] need not be 2-primal. For example considerZ2and let R=Z2⊕Z2. Then R is a commutative reduced ring with P(R) =0, and therefore R is 2-primal. Defineσ:R→R byσ(a,b) = (b,a). Then it can be seen that
P(R[x,x−1;σ]) =0, but P(R[x,x−1;σ])
is not completely semiprime as
((1, 0)x)2=0=P(R[x,x−1;σ]), but(1, 0)x ∈/ P(R[x,x−1;σ]).
Thus R[x,x−1;σ]is not 2-primal.
4. Skew Polynomial Rings Over
σ
(
∗
)
-rings
Letσbe an endomorphism of a ringRandδaσ-derivation ofRsuch that
σ(δ(a)) =δ(σ(a)) for all a∈R. Thenσ can be extended to an endomorphism (sayσ) of R[x;σ,δ]byσ(Pmi=0xiai) =
Pm i=0x
iσ(a
i). Alsoδcan be extended to aσ-derivation (sayδ)
ofR[x;σ,δ]byδ(Pm
i=0xiai) =
Pm
i=0xiδ(ai).
Example 5(Example 2.13 of Bhat[5]). Let R=R×R,σ:R→R defined byσ((a,b)) = (b,a) for a,b∈R. Thenσis an automorphism of R. Let now r∈R. Defineδr:R→R by
δr((a,b)) = (a,b)r−rσ((a,b))for a,b∈R. Thenδis aσ-derivation. Now for any(u,v)∈R, σ(δr((u,v))) =σ((u,v)r−rσ((u,v)))
=σ((u,v)r−r(v,u)) =σ((ur,v r)−σ(v r,ur)) =(v r,ur)−(ur,v r)).
Also
δr(σ((u,v))) =δr(v,u)
=(v,u)r−rσ((v,u)) =(v,u)r−r(u,v) =(v r,ur)−(ur,v r)).
Thereforeσ(δ((u,v))) =δ(σ((u,v)))for all(u,v)∈R.
Remark 3. We note that ifσ(δ(a))=6 δ(σ(a))for all a∈R, then the above does not hold. For example let f(x) =x l and g(x) =x p, a,b∈R. Then
δ(f(x)g(x)) =x2{δ(σ(l))σ(p) +σ(l)δ(p)}+x{δ2(l)σ(p) +δ(l)σ(p)},
but
δ(f(x))σ(g(x)) +f(x)δ(g(x)) = x2{σ(δ(l))σ(p) +σ(l)δ(p)}+x{δ2(l)σ(p) +δ(l)σ(p)}.
With this we now prove the following:
Theorem 3. Let R be a Noetherian ring which is also an algebra overQ. Letσbe an automor-phism of R andδ a σ-derivation of R such thatσ(δ(a)) =δ(σ(a))for all a ∈R. Further let P ∈ M in.S pec(O(R)) implies that P∩R ∈ M in.S pec(R). Then R is a σ(∗)-ring implies that O(R) =R[x;σ,δ]is a Noetherianσ(∗)-ring.
Proof. LetRbe a Noetherian ring andσan automorphism ofRsuch thatRis aσ(∗)-ring. We shall prove that O(R) =R[x;σ,δ] is a Noetherianσ(∗)-ring. For this we will show that any minimalP∈M in.S pec(O(R))is completely prime andσ(P) =P.
Let P ∈ M in.S pec(O(R)). Now P∩R ∈ M in.S pec(R) andR is a σ(∗)-ring implies that
σ(P∩R) = P∩R andP∩Ris a completely prime ideal ofR. Now Proposition (2.1) of Bhat [2]implies thatδ(P∩R)⊆P∩R. Now Theorem (2.4) of Bhat[4]implies thatO(P∩R)is a completely prime ideal ofO(R). NowO(P∩R)⊆Pimplies thatO(P∩R) =PasPis minimal. Nowσ(P∩R) =P∩Rimplies thatσ(P) =P.
Thusσ(P) =P andPis completely prime for allP∈M in.S pec(O(R)). Moreover
O(R) =R[x;σ,δ]is Noetherian by Theorem (1.12) of Goodearl and Warfield[7]. Hence by Proposition 2R[x;σ,δ]is aσ(∗)-ring.
We note that the condition that P ∈ M in.S pec(O(R))implies that P∩R ∈ M in.S pec(R) can not be ignored as follows:
LetR=Q×Q. Letσ:R→Rbe defined byσ((a,b)) = (b,a)andδ=0. Then P=0 is a prime ideal ofO(R), but P∩Ris not a prime ideal ofR.
We have not been able to prove the converse part of the above result. The main reason being that a generalization of Theorem 1 in terms ofO(R)is not known. The known towards this is:
LetRbe a Noetherian ring which is also an algebra overQ. Letσbe an automorphism ofR andδaσ-derivation ofR. ThenU ∈M in.S pec(R)such thatσ(U) =U implies thatδ(U)⊆U (Lemma 2.6 of Bhat[3]).
Question Let R be a Noetherian ring which is also an algebra over Q. Let σ be an automorphism ofRandδaσ-derivation ofR. IfO(R) =R[x;σ,δ]is a Noetherianσ(∗) -ring. IsRis aσ(∗)-ring?
References
[1] V. K. Bhat. Associated prime ideals of skew polynomial rings, Beitrage zur Algebra und Geometrie, Vol. 49(1). 277-283. 2008.
[2] V. K. Bhat. On Near Pseudo-valuation rings and their extensions, International Elec-tronic Journal of Algebra, 5:70-77, 2009.
[4] V. K. Bhat. A note on completely prime ideals of Ore extensions, International Jour-nal of Algebra and Computation, Vol. 20(3). 457-463. 2010.
[5] V. K. Bhat. Associated prime ideals of weakσ-rigid rings and their extensions, Alge-bra and Discrete Mathematics, Vol.10(1). 8-17. 2010.
[6] V. K. Bhat. Prime Ideals ofσ(∗)-Rings and their Extensions, Lobachevskii Journal of Mathematics, Vol. 32(1). 102-106. 2011.
[7] K. R. Goodearl and R. B. Warfield Jr. An introduction to non-commutative Noethe-rian rings, Cambridge University Press, 1989.
[8] J. Krempa. Some examples of reduced rings, Algebra Colloqium, Vol. 3(4). 289-300. 1996.
[9] T. K. Kwak. Prime radicals of skew-polynomial rings, International Journal of Math-ematical Sciences, Vol. 2(2). 219-227. 2003.
[10] G. Marks. On 2-primal Ore extensions, Communications in Algebra, Vol. 29(5). 2113-2123. 2001.
