On Pseudo-valuation rings and their extensions
Let R be a commutative Noetherian Q-algebra (Q is the field of rational numbers). Let σ be an automorphism of R and δ a σ-derivation of R. We define a δ-divided ring and prove the following: (1)If R is a pseudo-valuation ring such that x∉P for any prime ideal P of R[x;σ,δ], and P∩R is a prime...
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irk-123456789-1548622019-06-17T01:30:33Z On Pseudo-valuation rings and their extensions Bhat, V.K. Let R be a commutative Noetherian Q-algebra (Q is the field of rational numbers). Let σ be an automorphism of R and δ a σ-derivation of R. We define a δ-divided ring and prove the following: (1)If R is a pseudo-valuation ring such that x∉P for any prime ideal P of R[x;σ,δ], and P∩R is a prime ideal of R with σ(P∩R)=P∩R and δ(P∩R)⊆P∩R, then R[x;σ,δ] is also a pseudo-valuation ring. (2)If R is a δ-divided ring such that x∉P for any prime ideal P of R[x;σ,δ], and P∩R is a prime ideal of R with σ(P∩R)=P∩R and δ(P∩R)⊆P∩R, then R[x;σ,δ] is also a δ-divided ring. 2011 Article On Pseudo-valuation rings and their extensions / V.K. Bhat // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 2. — С. 25–30. — Бібліогр.: 14 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16S36, 16N40, 16P40, 16S32 http://dspace.nbuv.gov.ua/handle/123456789/154862 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let R be a commutative Noetherian Q-algebra (Q
is the field of rational numbers). Let σ be an automorphism of R and δ a σ-derivation of R. We define a δ-divided ring and prove the following:
(1)If R is a pseudo-valuation ring such that x∉P for any prime ideal P of R[x;σ,δ], and P∩R is a prime ideal of R with σ(P∩R)=P∩R and δ(P∩R)⊆P∩R, then R[x;σ,δ] is also a pseudo-valuation ring.
(2)If R is a δ-divided ring such that x∉P for any prime ideal P of R[x;σ,δ], and P∩R is a prime ideal of R with σ(P∩R)=P∩R and δ(P∩R)⊆P∩R, then R[x;σ,δ] is also a δ-divided ring. |
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Article |
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Bhat, V.K. |
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Bhat, V.K. On Pseudo-valuation rings and their extensions Algebra and Discrete Mathematics |
| author_facet |
Bhat, V.K. |
| author_sort |
Bhat, V.K. |
| title |
On Pseudo-valuation rings and their extensions |
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On Pseudo-valuation rings and their extensions |
| title_full |
On Pseudo-valuation rings and their extensions |
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On Pseudo-valuation rings and their extensions |
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On Pseudo-valuation rings and their extensions |
| title_sort |
on pseudo-valuation rings and their extensions |
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Інститут прикладної математики і механіки НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/154862 |
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On Pseudo-valuation rings and their extensions / V.K. Bhat // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 2. — С. 25–30. — Бібліогр.: 14 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT bhatvk onpseudovaluationringsandtheirextensions |
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2025-07-14T06:55:58Z |
| last_indexed |
2025-07-14T06:55:58Z |
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1837604440329158656 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 12 (2011). Number 2. pp. 25 – 30
c© Journal “Algebra and Discrete Mathematics”
On Pseudo-valuation rings and their extensions
V. K. Bhat1
Communicated by R. Wisbauer
Abstract. Let R be a commutative Noetherian Q-algebra
(Q is the field of rational numbers). Let σ be an automorphism of
R and δ a σ-derivation of R. We define a δ-divided ring and prove
the following:
(1) If R is a pseudo-valuation ring such that x /∈ P for any prime
ideal P of R[x;σ, δ], and P ∩ R is a prime ideal of R with
σ(P ∩ R) = P ∩ R and δ(P ∩ R) ⊆ P ∩ R, then R[x;σ, δ] is
also a pseudo-valuation ring.
(2) If R is a δ-divided ring such that x /∈ P for any prime ideal P of
R[x;σ, δ], and P∩R is a prime ideal of R with σ(P∩R) = P∩R
and δ(P ∩R) ⊆ P ∩R, then R[x;σ, δ] is also a δ-divided ring.
1. Introduction
Throughout this paper, all rings are associative with identity 1 6= 0.
Let now R be a ring. N(R) denotes the set of all nilpotent elements of
R. Z(R) denotes the center of R. Q denotes the field of rational numbers
and Z denotes the ring of integers unless otherwise stated. We recall that
as in Hedstrom [12], an integral domain R with quotient field F , is called
a pseudo-valuation domain (PVD) if each prime ideal P of R is strongly
prime (ab ∈ P , a ∈ F , b ∈ F implies that either a ∈ P or b ∈ P ). For
example let F = Q(
√
2). Set V = F + xF [[x]] = F [[x]]. Then V is a
pseudo-valuation domain. In Badawi, Anderson and Dobbs [4], the study
1The author would like to express his sincere thanks to the referee for his suggestions.
2000 Mathematics Subject Classification: 16S36, 16N40, 16P40, 16S32.
Key words and phrases: Automorphism, derivation, strongly prime ideal, divided
prime ideal, pseudo-valuation ring.
26 On Pseudo-valuation rings and their extensions
of pseudo-valuation domains was generalized to arbitrary rings in the
following way:
A prime ideal P of R is said to be strongly prime if aP and bR are
comparable (under inclusion; i.e. aP ⊆ bR or bR ⊆ aP ) for all a, b ∈ R.
The sets of prime ideals of R and strongly prime ideals of R are denoted
by Spec(R) and S.Spec(R) respectively.
A ring R is said to be a pseudo-valuation ring (PVR) if each prime
ideal P of R is strongly prime. We note that a PVR is quasilocal by
Lemma 1(b) of Badawi, Anderson and Dobbs [4].
An integral domain is a PVR if and only if it is a PVD by Proposition
(3.1) of Anderson [1], Proposition (4.2) of Anderson [2] and Proposition
(3) of Badawi [5]. We recall that a prime ideal P of R is said to be divided
if it is comparable (under inclusion) to every ideal of R. A ring R is called
a divided ring if every prime ideal of R is divided.
In Badawi [6], another generalization of PVDs is given in the following
way:
For a ring R with total quotient ring Q such that N(R) is a divided
prime ideal of R, let φ : Q → RN(R) such that φ(a/b) = a/b for every
a ∈ R and every b ∈ R\Z(R). Then φ is a ring homomorphism from Q
into RN(R), and φ restricted to R is also a ring homomorphism from R
into RN(R) given by φ(r) = r/1 for every r ∈ R. Denote RN(R) by T .
A prime ideal P of φ(R) is called a T -strongly prime ideal if xy ∈ P ,
x ∈ T , y ∈ T implies that either x ∈ P or y ∈ P . φ(R) is said to be a
T -pseudo-valuation ring (T -PVR) if each prime ideal of φ(R) is T-strongly
prime. A prime ideal S of R is called φ-strongly prime ideal if φ(S) is
a T -strongly prime ideal of φ(R). If each prime ideal of R is φ-strongly
prime, then R is called a φ-pseudo-valuation ring (φ− PV R).
This article is concerned with the study of skew polynomial rings
over PVDs. Let R be a ring and σ be an automorphism of R and δ
be a σ-derivation of R (δ : R → R is an additive map with δ(ab) =
δ(a)σ(b) + aδ(b), for all a, b ∈ R).
Example 1.1. Let σ be an automorphism of a ring R and δ : R → R
any map. Let φ : R → M2(R) defined by
φ(r) =
(
σ(r) 0
δ(r) r
)
, for all r ∈ R. Then δ is a σ-derivation of R if and
only if φ is a homomorphism.
We denote the Ore extension R[x;σ, δ] by O(R). If I is an ideal of R
such that I is σ-stable; i.e. σ(I) = I and I is δ-invariant; i.e. δ(I) ⊆ I, then
we denote I[x;σ, δ] by O(I). We would like to mention that R[x;σ, δ] is
the usual set of polynomials with coefficients in R, i.e. {
∑n
i=0 x
iai, ai ∈ R}
V. K. Bhat 27
in which multiplication is subject to the relation ax = xσ(a) + δ(a) for
all a ∈ R.
In case δ is the zero map, we denote the skew polynomial ring R[x;σ]
by S(R) and for any ideal I of R with σ(I) = I, we denote I[x;σ] by
S(I).
In case σ is the identity map, we denote the differential operator ring
R[x; δ] by D(R) and for any ideal J of R with δ(J) ⊆ J , we denote J [x; δ]
by D(J).
Ore-extensions including skew-polynomial rings and differential opera-
tor rings have been of interest to many authors. See [3, 7, 8, 9, 10, 13].
Polynomial rings over Pseudo-valuation rings
Recall that in Bhat [8], a prime ideal P of a ring R is σ-divided (σ
is an automorphism of R) if it is comparable (under inclusion) to every
σ-stable I of R. A ring R is called a σ-divided ring if every prime ideal
of R is σ-divided. In this direction in Theorem (2.8) of Bhat [8] it has
been proved that if R is a σ-divided Noetherian ring such that x /∈ P
for any P ∈ Spec(S(R)), then S(R) is also σ-divided Noetherian. Also in
Theorem (2.6) of Bhat [8] it has been proved that if R is a commutative
PVR such that x /∈ P for any P ∈ Spec(S(R)), then S(R) is also a PVR.
In this paper, we generalize these results for O(R) and answer Question
(1) of [8], but before that we have the following:
Let R be a ring. Let σ be an automorphism of R and δ a σ-derivation
of R. We say that a prime ideal P of R is δ-divided if it is comparable
(under inclusion) to every σ-stable and δ-invariant ideal I of R. A ring R
is called a δ-divided ring if every prime ideal of R is δ-divided.
Let now R be a commutative Noetherian Q-algebra. Let σ be an
automorphism of R and δ a σ-derivation of R. Then we prove the following:
(1) Let R be a pseudo-valuation ring such that x /∈ P for any P ∈
Spec(O(R)) and P ∩R is a σ-stable and δ-invariant prime ideal of
R. Further assume that for any U ∈ S.Spec(R) with σ(U) = U and
δ(U) ⊆ U we have O(U) ∈ S.Spec(O(R)). Then R[x;σ, δ] is also a
pseudo-valuation ring.
(2) Let R be a δ-divided ring such that x /∈ P for any P ∈ Spec(O(R))
and P ∩ R be a σ-stable and δ-invariant prime ideal of R. Then
R[x;σ, δ] is also a δ-divided ring.
These results are proved in Theorems (2.3) and (2.8) respectively.
28 On Pseudo-valuation rings and their extensions
2. Polynomial rings
We begin with the following known results:
Lemma 2.1. Let R be a commutative Noetherian Q-algebra. Let σ be an
automorphism of R and δ be a σ-derivation of R. Then U is a prime ideal
of R such that σ(U) = U and δ(U) ⊆ U implies that O(U) is a prime
ideal of O(R) and O(U) ∩R = U .
Proof. The proof is on the same lines as in Theorem (2.22) of Goodearl
and Warfield [11] and Lemma (10.6.4) of McConnell and Robson [14].
Theorem 2.2. (Hilbert Basis Theorem): Let R be a right/left Noetherian
ring. Let σ be an automorphism of R and δ a σ-derivation of R. Then the
Ore extension O(R) = R[x;σ, δ] is right/left Noetherian.
Proof. See Theorem (1.12) of Goodearl and Warfield [11].
Theorem 2.3. Let R be a commutative Noetherian pseudo-valuation
Q-algebra such that x /∈ P for any P ∈ Spec(O(R)) and P ∩ R be a
σ-stable and δ-invariant prime ideal of R. Further let any U ∈ S.Spec(R)
with σ(U) = U and δ(U) ⊆ U implies that O(U) ∈ S.Spec(O(R)). Then
O(R) is a Noetherian pseudo-valuation Q-algebra.
Proof. O(R) is Noetherian by Theorem (2.2). Let J ∈ Spec(O(R)). Then
J ∩R ∈ Spec(R) with σ(J ∩R) = J ∩R and δ(J ∩R) ⊆ J ∩R. Now R
is a pseudo-valuation Q-algebra, therefore J ∩ R ∈ S.Spec(R). Now by
hypothesis O(J∩R) ∈ S.Spec(O(R)). Now it can be seen that O(J∩R) =
J . Therefore J ∈ S.Spec(O(R)). Hence O(R) is a pseudo-valuation Q-
algebra.
Corollary 2.4. Let R be a PVR such that x /∈ P for any P ∈ Spec(S(R)).
Then S(R) is also a PVR.
We note that Theorem (2.3) does not hold without the condition that
P ∩R is a σ-stable and δ-invariant prime ideal of R.
Example 2.5. Let R = Q×Q. Let σ : R 7→ R be defined by σ
(
(a, b)
)
=
(b, a), and δ = 0. Then P = 0 is a prime ideal of O(R) such that x /∈ P ,
but P ∩R is not a prime ideal of R.
Now let R and σ be as above, and δ = Id - σ. Then δ is a σ-derivation
of R. Now it can be seen that O(R) has the form R[x − 1;σ]. Now
P = (1, 0)R+ (x− 1)O(R) is a prime ideal of O(R) such that x /∈ P , but
P ∩R = (1, 0)R is not σ-stable or δ-invariant.
V. K. Bhat 29
We also note that in Theorem (2.3) the hypothesis that any U ∈
S.Spec(R) with σ(U) = U and δ(U) ⊆ U implies that O(U) ∈
S.Spec(O(R)) can not be deleted as an extension of a strongly prime
ideal of R need not be a strongly prime ideal of O(R).
Example 2.6. R = Z(p). This is in fact a discrete valuation domain, and
therefore, its maximal ideal P = pR is strongly prime. But pR[x] is not
strongly prime in R[x] because it is not comparable with xR[x] (so the
condition of being strongly prime in R[x] fails for a = 1 and b = x).
Corollary 2.7. Let R be a commutative Noetherian pseudo-valuation
Q-algebra such that x /∈ P for any P ∈ Spec(D(R)). Then D(R) is a
Noetherian pseudo-valuation Q-algebra.
We note that Corollary (2.7) does not hold without the condition
that x /∈ P for any P ∈ Spec(D(R)). For example let R = Q[y](y)
(
the
localization of the polynomial ring Q[y] at the maximal ideal (y)
)
and
δ = y d
dy
. Then R is a commutative PVR. Now P = yD(R) + xD(R) is
a prime (maximal) ideal of D(R), but xP is not comparable to yD(R),
therefore D(R) is not a PVR.
Theorem 2.8. If R is a δ-divided commutative Noetherian Q-algebra
such that x /∈ P for any P ∈ Spec(O(R)) and P ∩ R is a σ-stable and
δ-invariant prime ideal of R, then O(R) is δ-divided Noetherian Q-algebra.
Proof. O(R) is Noetherian by Theorem (2.2). Let J ∈ Spec(O(R)) and
0 6= K be a proper ideal of O(R) such that σ(K) = K and δ(K) ⊆ K.
Now we note that σ can be extended to an automorphism of O(R) such
that σ(x) = x and δ can be extended to a σ-derivation of O(R) such that
δ(x) = 0. Now J∩R ∈ Spec(R) with σ(J∩R) = J∩R and δ(J∩R) ⊆ J∩R.
Also K∩R is an ideal of R with σ(K∩R) = K∩R and δ(K∩R) ⊆ K∩R.
Now R is divided, therefore J∩R and K∩R are comparable under inclusion.
Say J ∩R ⊆ K∩R. Therefore O(J ∩R) ⊆ O(K∩R). Thus J ⊆ K. Hence
O(R) is δ-divided Noetherian.
Corollary 2.9. Let R be a σ-divided Noetherian ring such that x /∈ P
for any P ∈ Spec(S(R)). Then S(R) is also σ-divided Noetherian.
Corollary 2.10. Let R be a divided commutative Noetherian Q-algebra
such that x /∈ P for any P ∈ Spec(D(R)). Then D(R) is also divided
Noetherian.
We note that Corollary (2.10) does not hold without the condition
that x /∈ P for any P ∈ Spec(D(R)). For example let R = Q[y](y)
(
the
localization of the polynomial ring Q[y] at the maximal ideal (y)
)
and
30 On Pseudo-valuation rings and their extensions
δ = y d
dy
. Then R is a commutative PVR, and so it is a divided ring. Now
P = yD(R) is a prime ideal of D(R), but it is not comparable to the ideal
y2D(R) + xD(R), and therefore D(R) is not divided.
Question 2.11. (Question 1 of [8]): Let R be a PVR. Let σ be an
automorphism of R and δ a σ-derivation of R. Is O(R) = R[x;σ, δ] a PVR
(even if R is commutative Noetherian)?
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(1979), 451-463.
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325-332.
[3] S. Annin, Associated primes over skew polynomial rings, Commun. Algebra 30
(2002), 2511-2528.
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Notes Pure Appl. Math. 185 (1997), 57-67, Marcel Dekker, New York.
[5] A. Badawi, On domains which have prime ideals that are linearly ordered, Commun.
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(1999), 101-110, Marcel Dekker, New York.
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Math. 22 (2006), 3-6.
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Sci. (2007), Art. ID 20138, 6 pages, doi: 10.1155.
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into artinian rings, Proc. Amer. Math. Soc. 109(4) (1990), 881-886.
[11] K. R. Goodearl and R. B. Warfield Jr, An introduction to non-commutative
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(1978), 551-567.
[13] C. Y. Hong, N. K. Kim and T. K. Kwak, Ore-extensions of Baer and p.p.-rings, J.
Pure and Applied Algebra. 151(3) (2000), 215-226.
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ley(1987); revised edition: Amer. Math. Soc. (2001).
Contact information
V. K. Bhat School of Mathematics, SMVD University, P/o
SMVD University, Katra, J and K, India- 182320
E-Mail: vijaykumarbhat2000@yahoo.com
Received by the editors: 14.03.2011
and in final form 14.03.2011.
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