Volume 2007, Article ID 20138,6pages doi:10.1155/2007/20138
Research Article
Polynomial Rings over Pseudovaluation Rings
V. K. Bhat
Received 5 March 2007; Accepted 1 August 2007
Recommended by Howard E. Bell
LetRbe a ring. Letσ be an automorphism ofR. We define aσ-divided ring and prove the following. (1) LetRbe a commutative pseudovaluation ring such thatx∈Pfor any P∈Spec(R[x,σ]) . ThenR[x,σ] is also a pseudovaluation ring. (2) LetRbe aσ-divided ring such thatx∈P for anyP∈Spec(R[x,σ]). ThenR[x,σ] is also aσ-divided ring. Let nowRbe a commutative NoetherianQ-algebra (Qis the field of rational numbers). Let δ be a derivation ofR. Then we prove the following. (1) LetR be a commutative pseudovaluation ring. ThenR[x,δ] is also a pseudovaluation ring. (2) LetRbe a divided ring. ThenR[x,δ] is also a divided ring.
Copyright © 2007 V. K. Bhat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and repro-duction in any medium, provided the original work is properly cited.
1. Introduction
All rings are associative with identity 1. Now letRbe a ring.N(R) denotes the set of all nilpotent elements ofR.Z(R) denotes the centre of R.Qdenotes the field of rational numbers unless otherwise stated. We recall that as in Hedstrom and Houston [1], an integral domainRwith quotient fieldF, is called a pseudovaluation domain (PVD) if each prime idealPofRis strongly prime (ab∈P,a∈F,b∈Fimplies that eithera∈P orb∈P). In Badawi et al. [2], the study of pseudovaluation domains was generalized to arbitrary rings in the following way.
A prime idealPofRis said to be strongly prime ifaPandbRare comparable (under inclusion) for all a,b∈R. A commutative ringR is said to be a pseudovaluation ring (PVR) if each prime idealPofRis strongly prime. We note that a commutative PVR is quasilocal by Badawi et al. [2, Lemma 1(b)].
PofRis said to be divided if it is comparable (under inclusion) to every ideal ofR. A ring Ris called a divided ring if every prime ideal ofRis divided. We denote the set of prime ideals ofRby Spec(R) and the set of strongly prime ideals ofRbyS·Spec(R).
In Badawi [6], another generalization of PVDs is given in the following way:
For a ringRwith total quotient ringQsuch thatN(R) is a divided prime ideal ofR, letφ:Q→RN(R)such thatφ(a/b)=a/b for everya∈Rand everyb∈R\Z(R). Thenφ is a ring homomorphism fromQintoRN(R), andφrestricted toRis also a ring homo-morphism fromRintoRN(R)given byφ(r)=r/1 for everyr∈R. DenoteRN(R)byT. A prime idealPofφ(R) is called aT-strongly prime ideal ifxy∈P,x∈T,y∈Timplies that eitherx∈P ory∈P.φ(R) is said to be aT-pseudovaluation ring (T-PVR) if each prime ideal ofφ(R) isT-strongly prime. A prime idealSofRis calledφ-strongly prime ideal ifφ(S) is aT-strongly prime ideal ofφ(R). If each prime ideal ofRisφ-strongly prime, thenRis called aφ-pseudovaluation ring (φ-PVR).
Also recall from Badawi [7], a ringRis called aφ-chained ring (φ-CR) ifN(R) is a divided prime ideal ofRand for everya∈T\φ(R), we havea−1∈φ(R). In Badawi [8, Proposition 2.6], it is shown that ifN(R) is a divided prime ideal ofR, andPis a regular φ-strongly prime ideal ofR. Then the total quotient ringQofRisφ-CR.
This article concerns the study of skew polynomial rings over PVDs. LetRbe a ring andσbe an automorphism ofR. We denote the skew polynomial ringR[x,σ] byS(R). If Iis an ideal ofRsuch thatIisσ-stable; that is,σ(I)=I, then we denoteI[x,σ] byS(I). We would like to mention thatR[x,σ] is the usual set of polynomials with coefficients in R, that is,{ni=0xiai,ai∈R}in which multiplication is subject to the relationax=xσ(a) for alla∈R.
LetRbe a ring andσbe an automorphism ofR. We denote the skew Laurent polyno-mial ringR[x,x−1,σ] byL(R). We would also like to mention thatL(R)= {n
i=−mxiai,ai∈
R}in which multiplication is subject to the relationax=xσ(a) for alla∈R. IfIis an ideal ofRsuch thatσ(I)=I, then we denoteI[x,x−1,σ] byL(I).
LetRbe a ring andδbe a derivation ofR. We denote the differential operator ring R[x,δ] byD(R). IfIis an ideal ofRsuch thatδ(I)⊆I, then we denoteI[x,δ] byD(I). We would like to mention thatD(R) is the usual set of polynomials with coefficients inR, that is,{ni=0xiai,ai∈R}in which multiplication is subject to the relationax=xa+δ(a) for alla∈R.
Ore-extensions including skew polynomial rings and differential operator rings have been of interest to many authors. See [9–12].
We define aσ-divided ring (σis an automorphism ofR) in the following way.
LetRbe a ring. We say that a prime idealPofRisσ-divided if it is comparable (under inclusion) to everyσ-stable idealIofR. A ringRis called aσ-divided ring if every prime ideal ofRisσ-divided.
Let nowRbe a ring. Letσbe an automorphism ofR. Then we prove the following. (1) Let R be a commutative pseudovaluation ring such that x /∈P for any P∈
Spec(S(R)). ThenR[x,σ] is also a pseudovaluation ring.
(2) LetRbe aσ-divided ring such thatx /∈Pfor anyP∈Spec(S(R)). ThenR[x,σ] is also aσ-divided ring.
Let nowRbe a commutative NoetherianQ-algebra. Letδbe a derivation ofR. Then we prove the following.
(1) LetRbe a commutative pseudovaluation ring. ThenR[x,δ] is also a pseudoval-uation ring.
(2) LetRbe a divided ring. ThenR[x,δ] is also a divided ring. These results are proved in Theorems2.10and2.11, respectively.
2. Polynomial rings
We begin with the following known results.
Lemma 2.1. LetRbe a ring. Letσbe an automorphism ofR.
(1) IfPis a prime ideal ofS(R) such thatx /∈P, thenP∩Ris a prime ideal ofRand σ(P∩R)=P∩R.
(2) IfQis a prime ideal ofRsuch thatσ(Q)=Q, thenS(Q) is a prime ideal ofS(R)
andS(Q)∩R=Q.
Proof. The proof follows on the same lines as in McConnell and Robson [13,14, Lemma
10.6.4].
Lemma 2.2. LetRbe a commutative NoetherianQ-algebra. Letδbe a derivation ofR. Then:
(1) IfPis a prime ideal ofD(R), thenP∩Ris a prime ideal ofRandδ(P∩R)⊆P∩R.
(2) IfUis a prime ideal ofRsuch thatδ(U)⊆U, thenD(U) is a prime ideal ofD(R)
andD(U)∩R=U.
Proof. See Goodearl and Warfield [15, Theorem 2.22].
Lemma 2.3. LetRbe a Noetherian ring. Letσbe an automorphism ofR. IfIis a prime ideal ofRsuch thatσ(I)⊆I, thenL(I) is an ideal ofL(R) and ifJis an ideal ofL(R), thenJ∩R is an ideal ofRandσ(J∩R)⊆J∩R.
Proof. See Goodearl and Warfield [15, Example 2ZA].
LetRbe a ring. Letαbe an automorphism ofRandρbe anα-derivation ofR, that is, ρ(ab)=ρ(a)α(b) +aρ(b), fora,b∈R. Then Ore-extensionR[x,α,ρ] is the usual set of polynomials with coefficients inR, that is,{ni=0xiai,ai∈R}in which multiplication is subject to the relationax=xα(a) +ρ(a) for alla∈R.
Theorem 2.4 (Hilbert Basis theorem). LetRbe a right/left Noetherian ring. Letαandρbe as above. Then the ore-extensionO(R)=R[x,α,ρ] is right/left Noetherian. AlsoR[x,x−1,α] is right/left Noetherian.
Proof. See Goodearl and Warfield [15, Theorems 1.12 and 1.17].
Proposition 2.5. LetRbe a ring. Letσbe an automorphism ofRandδbe aσ-derivation ofR. Then the following hold.
(1) For any strongly prime idealPofRwithδ(P)⊆Pandσ(P)=P,O(P)=P[x,σ,δ]
is a strongly prime ideal ofO(R).
Proof. (1) LetP be a strongly prime ideal of R. Now let f(x)=in=0xiai∈O(R) and g(x)=mj=0xjbj∈O(R) be such that f(x)g(x)∈O(P). Suppose f(x)∈/ O(P). We will show thatg(x)∈O(P). We use induction onnandm. Forn=m=1, the verification is easy. We check forn=2 andm=1. Let f(x)=x2a+xb+candg(x)=xu+v. Now f(x)g(x)∈O(P) with f(x)∈/ O(P). The possibilities area /∈Porb /∈Porc /∈Por any two out of these three do not belong toPor all of them do not belong toP. We verify case by case.
Let a /∈P. Sincex3σ(a)u+x2(δ(a)u+σ(b)u+av) +x(δ(b)u+σ(c)u+bv) +δ(c)u+ cv∈O(P), we haveσ(a)u∈P, and sou∈P. Nowδ(a)u+σ(b)u+av∈Pimpliesav∈P, and sov∈P. Therefore,g(x)∈O(P).
Letb /∈P. Nowσ(a)u∈P. Supposeu /∈P, thenσ(a)∈Pand thereforea,δ(a)∈P. Nowδ(a)u+σ(b)u+av∈P implies thatσ(b)u∈P which in turn implies thatb∈P, which is not the case. Therefore, we haveu∈P. Nowδ(b)u+σ(c)u+bv∈Pimplies that bv∈Pand thereforev∈P. Thus, we haveg(x)∈O(P).
Letc /∈P. Nowσ(a)u∈P. Supposeu /∈P, then as abovea,δ(a)∈P. Nowδ(a)u+ σ(b)u+av∈Pimplies thatσ(b)u∈P. Nowu /∈Pimplies thatσ(b)∈P; that is,b,δ(b)∈
P. Alsoδ(b)u+σ(c)u+bv∈Pimpliesσ(c)u∈Pand thereforeσ(c)∈Pwhich is not the case. Thus, we haveu∈P. Nowδ(c)u+cv∈Pimpliescv∈P, and sov∈P. Therefore, g(x)∈O(P).
Now suppose that the result is true for k,n=k >2 and m=1. We will prove for n=k+ 1. Let f(x)=xk+1a
k+1+xkak+···xa1+a0, and g(x)=xb1+b0 be such that
f(x)g(x)∈O(P), but f(x)∈/ O(P). We will show thatg(x)∈O(P). If ak+1∈/ P, then equating coefficients ofxk+2, we getσ(a
k+1)b1∈P, which implies thatb1∈P. Now equat-ing coefficients ofxk+1, we getσ(a
k)b1+ak+1b0∈P, which implies thatak+1b0∈P, and
thereforeb0∈P. Henceg(x)∈O(P).
Ifaj∈/ P, 0≤j≤k, then using induction hypothesis, we get thatg(x)∈O(P).
There-fore, the statement is true for alln. Now using the same process, it can be easily seen that the statement is true for allmalso. We leave the details to the reader.
(2) LetUbe a strongly prime ideal ofO(R). Supposea,b∈Rare such thatab∈(U∩
R) witha /∈(U∩R). This means thata /∈Uasa∈R. Thus we haveab∈(U∩R)⊆U, witha /∈U. Therefore, we haveb∈U, and thusb∈(U∩R).
Theorem 2.6. LetRbe a commutative PVR such thatx /∈Pfor anyP∈Spec(S(R)). Then S(R) is also a PVR.
Proof. LetJ∈Spec(S(R)). Then by Lemma 2.1,J∩R∈Spec(R) andσ(J∩R)=J∩R. NowRis a commutative PVR, thereforeJ∩R∈S·Spec(R). NowProposition 2.5implies that S(J∩R)∈S·Spec(D(R)). Now it is easy to see that S(J∩R)=J. Therefore,J∈ S·Spec(D(R)). Hence,S(R) is a PVR.
Corollary 2.7. LetRbe a commutative Noetherian ring which is also a PVR andσ(P)=P for allP∈Spec(R). ThenL(R) is also a PVR.
Proof. UseProposition 2.5and Goodearl and Warfield [15, Example 2ZA].
Theorem 2.8. LetRbe aσ-divided Noetherian ring such thatx /∈Pfor anyP∈Spec(S(R)).
Proof. We note thatσ can be extended to an automorphism ofS(R) such thatσ(x)=x. AlsoS(R) is Noetherian byTheorem 2.4. LetJ∈Spec(S(R)) and 0=Kbe a proper ideal of S(R) such thatσ(K)=K. Now by McConnell and Robson [13,14, Lemma 10.6.4], J∩R∈Spec(R) andσ(J∩R)=(J∩R). Also by McConnell and Robson [13,14, Lemma 10.6.3],K∩Ris an ideal ofRandσ(K∩R)=(K∩R). NowRisσ-divided, thereforeJ∩R andK∩Rare comparable under inclusion. Say (J∩R)⊆(K∩R). Therefore,S(J∩R)⊆ S(K∩R). ThusJ⊆K. Hence,S(R) isσ-divided Noetherian.
Corollary 2.9. LetRbe a divided Noetherian ring andσ(P)=Pfor allP∈Spec(R). Then L(R) is also divided.
Proof. Use Goodearl and Warfield [15, Example 2ZA].
Theorem 2.10. LetRbe a commutative NoetherianQ-algebra which is also a PVR. Then D(R) is also a PVR.
Proof. LetJ∈Spec(D(R)). Then byLemma 2.2,J∩R∈Spec(R) andδ(J∩R)⊆J∩R. NowRis a PVR, thereforeJ∩R∈S·Spec(R). NowProposition 2.5implies thatD(J∩
R)∈S·Spec(D(R)); but D(J∩R)=J by Lemma 2.2. Therefore, J∈S·Spec(D(R)).
HenceD(R) is a PVR.
Theorem 2.11. LetRbe a divided commutative NoetherianQ-algebra. ThenD(R) is also
divided Noetherian.
Proof. D(R) is Noetherian byTheorem 2.4. LetJ∈Spec(D(R)) and 0=K be a proper ideal ofD(R). Now by Goodearl and Warfield [15, Theorem 2.22],J∩R∈Spec(R) and δ(J∩R)⊆(J∩R). AlsoK∩Ris an ideal ofRandδ(K∩R)⊆(K∩R) by Goodearl and Warfield [15, Lemma 2.18]. NowRis divided, thereforeJ∩RandK∩Rare comparable under inclusion. Say (J∩R)⊆(K∩R). Therefore,D(J∩R)⊆D(K∩R). Thus, J⊆K. Hence,D(R) is divided Noetherian.
Question 1. LetRbe a commutative PVR. Letσ be an automorphism ofRandδ be a σ-derivation ofR. IsO(R)=R[x,σ,δ] a PVR (even ifRis Noetharian)?
References
[1] J. R. Hedstrom and E. G. Houston, “Pseudo-valuation domains,” Pacific Journal of Mathematics, vol. 75, no. 1, pp. 137–147, 1978.
[2] A. Badawi, D. F. Anderson, and D. E. Dobbs, “Pseudo-valuation rings,” in Commutative Ring Theory (F`es, 1995), vol. 185 of Lecture Notes in Pure and Appl. Math., pp. 57–67, Marcel Dekker, New York, NY, USA, 1997.
[3] D. F. Anderson, “Comparability of ideals and valuation overrings,” Houston Journal of Mathe-matics, vol. 5, no. 4, pp. 451–463, 1979.
[4] D. F. Anderson, “When the dual of an ideal is a ring,” Houston Journal of Mathematics, vol. 9, no. 3, pp. 325–332, 1983.
[5] A. Badawi, “On domains which have prime ideals that are linearly ordered,” Communications in Algebra, vol. 23, no. 12, pp. 4365–4373, 1995.
[7] A. Badawi, “Onφ-chained rings andφ-pseudo-valuation rings,” Houston Journal of Mathemat-ics, vol. 27, no. 4, pp. 725–736, 2001.
[8] A. Badawi, “On the complete integral closure of rings that admit aφ-strongly prime ideal,” in Commutative Ring Theory and Applications (Fez, 2001), vol. 231 of Lecture Notes in Pure and Appl. Math., pp. 15–22, Marcel Dekker, New York, NY, USA, 2003.
[9] S. Annin, “Associated primes over skew polynomial rings,” Communications in Algebra, vol. 30, no. 5, pp. 2511–2528, 2002.
[10] W. D. Blair and L. W. Small, “Embedding differential and skew polynomial rings into Artinian rings,” Proceedings of the American Mathematical Society, vol. 109, no. 4, pp. 881–886, 1990. [11] 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.
[12] T. K. Kwak, “Prime radicals of skew polynomial rings,” International Journal of Mathematical Sciences, vol. 2, no. 2, pp. 219–227, 2003.
[13] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Pure and Applied Math-ematics, John Wiley & Sons, Chichester, UK, 1987.
[14] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, vol. 30 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, revised edition, 2001. [15] K. R. Goodearl and R. B. Warfield Jr., An Introduction to Noncommutative Noetherian Rings,
vol. 16 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, UK, 1989.
V. K. Bhat: School of Applied Physics and Mathematics, Shri Mata Vaishno Devi University, P/o Kakryal, Katra 182301, India
