Differential operator rings over 2-primal rings
Let R be a ring, and δ be a derivation of R. It is proved that R is a 2-primal Noetherian Q-algebra implies that the differential operator ring R[x, δ] is a 2-primal Noetherian.
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Інститут прикладної математики і механіки НАН України
2008
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| Цитувати: | Differential operator rings over 2-primal rings / V.K. Bhat // Український математичний вісник. — 2008. — Т. 5, № 2. — С. 153-158. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-1243332017-09-24T03:03:55Z Differential operator rings over 2-primal rings Bhat, V.K. Let R be a ring, and δ be a derivation of R. It is proved that R is a 2-primal Noetherian Q-algebra implies that the differential operator ring R[x, δ] is a 2-primal Noetherian. 2008 Article Differential operator rings over 2-primal rings / V.K. Bhat // Український математичний вісник. — 2008. — Т. 5, № 2. — С. 153-158. — Бібліогр.: 11 назв. — англ. 1810-3200 2000 MSC. 16XX, 16N40, 16P40, 16W20, 16W25. http://dspace.nbuv.gov.ua/handle/123456789/124333 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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Let R be a ring, and δ be a derivation of R. It is proved that R is a 2-primal Noetherian Q-algebra implies that the differential operator ring R[x, δ] is a 2-primal Noetherian. |
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Bhat, V.K. |
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Bhat, V.K. Differential operator rings over 2-primal rings Український математичний вісник |
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Bhat, V.K. |
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Bhat, V.K. |
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Differential operator rings over 2-primal rings |
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Differential operator rings over 2-primal rings |
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Differential operator rings over 2-primal rings |
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Differential operator rings over 2-primal rings |
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Differential operator rings over 2-primal rings |
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differential operator rings over 2-primal rings |
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Інститут прикладної математики і механіки НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/124333 |
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Differential operator rings over 2-primal rings / V.K. Bhat // Український математичний вісник. — 2008. — Т. 5, № 2. — С. 153-158. — Бібліогр.: 11 назв. — англ. |
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Український математичний вісник |
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AT bhatvk differentialoperatorringsover2primalrings |
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2025-07-09T01:16:36Z |
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2025-07-09T01:16:36Z |
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1837130106202488832 |
| fulltext |
Український математичний вiсник
Том 5 (2008), № 2, 153 – 158
Differential operator rings over 2-primal rings
Vijay K. Bhat
(Presented by B. V. Novikov)
Abstract. Let R be a ring, and δ be a derivation of R. It is proved
that R is a 2-primal Noetherian Q-algebra implies that the differential
operator ring R[x, δ] is a 2-primal Noetherian.
2000 MSC. 16XX, 16N40, 16P40, 16W20, 16W25.
Key words and phrases. 2-primal, Minimal prime, prime radical, nil
radical, derivation.
Introduction
A ring R always means an associative ring with identity. Q denotes
the field of rational numbers. Spec(R) denotes the set of prime ideals
of R. MinSpec(R) denotes the sets of minimal prime ideals of R. P (R)
and N(R) denote the prime radical and the set of nilpotent elements of
R respectively. Let I and J be any two ideals of a ring R. Then I ⊂ J
means that I is strictly contained in J .
Before we discuss 2-primal rings, let us briefly recall the definitions of
some Ore extensions concerning this paper.
Recall that a derivation of a ring R is an additive map δ : R → R such
that δ(ab) = δ(a)b + aδ(b), for all a, b ∈ R. For example let R = K[x], K
is a field. Then the formal derivative d/dx is a derivation of R.
Differential operator ring R[x, δ] is the usual polynomial ring with
coefficients in R in which multiplication is subject to the relation ax =
xa + δ(a) for all a ∈ R. We take any f(x) ∈ R[x, δ] to be of the form
f(x) =
∑
n
i=0 xiai. We denote R[x, δ] by D(R). If I is a δ-invariant (i.e.
δ(I) ⊆ I) ideal of R, then I[x, δ] is an ideal of D(R). We denote I[x, δ]
as usual by D(I).
Received 21.03.2007
ISSN 1810 – 3200. c© Iнститут математики НАН України
154 Differential operator...
Let σ be an endomorphism of a ring R. A σ-derivation of R is an
additive map δ : R → R such that δ(ab) = δ(a)σ(b) + aδ(b), for all
a, b ∈ R. For example for any endomorphism τ of a ring R and for any
a ∈ R, ̺ : R → R defined as ̺(r) = ra − aτ(r) is a τ -derivation of R.
Also let σ be an automorphism of a ring R and δ : R → R any map. Let
φ : R → M2(R) be a map defined by
φ(r) =
(
σ(r) 0
δ(r) r
)
,
for all r ∈ R. Then δ is a σ-derivation of R.
Recall that the Ore extension R[x, σ, δ] is the usual polynomial ring
with coefficients in R in which multiplication is subject to the relation
ax = xσ(a) + δ(a) for all a ∈ R. We take any f(x) ∈ R[x, σ, δ] to be
of the form f(x) =
∑
n
i=0 xiai. We denote R[x, σ, δ] by O(R). If J is a
σ-stable (i.e. σ(J) = J) ideal and δ-invariant (i.e. δ(J) ⊆ J) ideal of R,
then J [x, σ, δ] is an ideal of O(R). We denote J [x, σ, δ] as usual by O(J).
Now this article concerns the study of differential operator rings in
terms of 2-primal rings. We recall that a ring R is 2-primal if and only if
the set of nilpotent elements of R and the prime radical of R are same if
and only if the prime radical is a completely semiprime ideal. An ideal I
of a ring R is called completely semiprime if a2 ∈ I implies a ∈ I, where
a ∈ R. Also I is called completely prime if ab ∈ I implies a ∈ I or b ∈ I
for a, b ∈ R. We note that a reduced ring is 2-primal and a commutative
ring is also 2-primal. For further details on 2-primal rings, we refer the
reader to [2, 4, 7, 9, 10].
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 of R[x, σ, δ], where R is a local ring, σ is
an automorphism of R and δ is a σ-derivation of R.
Minimal prime ideals of 2-primal rings have been discussed by Kim
and Kwak in [9]. 2-primal near rings have been discussed by Argac and
Groenewald in [2].
Let R be a ring, σ be an automorphism of R and δ be a σ-derivation of
R. Recall that as defined in [4] a ring R is called a δ-ring if aδ(a) ∈ P (R)
implies a ∈ P (R), where P (R) denotes the prime radical of R. Note that
a ring with identity is not a δ-ring. The following result has been proved
in Theorem 2.8 of [4] concerning δ-rings:
Let R be a δ-Noetherian Q-algebra such that σ(δ(a)) = δ(σ(a)), for
all a ∈ R; σ(P ) = P for all P ∈ MinSpec(R) and δ(P (R)) ⊆ P (R). Then
R[x, σ, δ] is 2-primal.
V. K. Bhat 155
Now there arises a natural question:
Let R be a 2-primal ring. Is R[x, σ, δ] also a 2-primal ring? For the
time being we are not able to answer this question, but towards this we
prove the following result in this paper:
Let R be a 2-primal Noetherian Q-algebra. Then R[x, δ] is 2-primal
Noetherian. This is proved in Theorem 1.2.
Before proving the above result, we find a relation between the min-
imal prime ideals of R and those of R[x, δ], where R is a Noetherian
Q-algebra. This is proved in Theorem 1.1.
Ore-extensions including skew-polynomial rings and differential oper-
ator rings have been of interest to many authors. See [1, 3, 4, 8].
1. Differential operator ring D(R) = R[x, δ]
Before we proceed further, we recall that Gabriel proved in Lemma
3.4 of [5] that if R is a Noetherian Q-algebra and δ is a derivation of R,
then δ(P ) ⊆ P , for all P ∈ MinSpec(R). This result has been generalized
in Theorem 2.2 of [4] for a σ-derivation δ of R and it has been proved
that if R is a Noetherian Q-algebra. If σ is an automorphism of R and
δ is a σ-derivation of R such that σ(δ(a)) = δ(σ(a)), for all a ∈ R, then
any P ∈ MinSpec(O(R)) with σ(P ) = P implies that δ(P ) ⊆ P .
The following Proposition follows immediately from Theorem 2.2 of
[4], but we give a sketch of the proof in order to make the paper self
contained.
Proposition 1.1. Let R be a Noetherian Q-algebra. Let δ be a derivation
of R. Then δ(P (R)) ⊆ P (R).
Proof. Let P1 ∈ MinSpec(R). Let T = R[[t]], the formal power se-
ries ring. Now it can be seen that etδ is an automorphism of T and
P1T ∈ MinSpec(T ). We also know that (etδ)k(P1T ) ∈ MinSpec(T ) for
all integers k ≥ 1. Now T is Noetherian by Exercise 1ZA(c) of Goodearl
and Warfield [6], and therefore Theorem 2.4 of Goodearl and Warfield [6]
implies that MinSpec(T ) is finite. So exists an integer an integer n ≥ 1
such that (etδ)n(P1T ) = P1T ; i.e. (entδ)(P1T ) = P1T . But R is a Q-
algebra, therefore, etδ(P1T ) = P1T . Now for any a ∈ P1, a ∈ P1T also,
and so etδ(a) ∈ P1T ; i.e. a + tδ(a) + (t2/2!)δ2(a) + · · · ∈ P1T , which
implies that δ(a) ∈ P1. Therefore δ(P1) ⊆ P1.
Now P (R) ⊆ P , for all P ∈ MinSpec(R) implies that δ(P (R)) ⊆
δ(P ) ⊆ P , for all P ∈ MinSpec(R). Therefore
δ(P (R)) ⊆ ∩P∈MinSpec(R)P = P (R).
156 Differential operator...
Proposition 1.2. Let R be a Noetherian Q-algebra. Let δ be as usual.
Then D(N(R)) = N(D(R)).
Proof. It is easy to see that D(N(R)) ⊆ N(D(R)). We will show that
N(D(R)) ⊆ D(N(R)).
Let f =
∑
m
i=0 xiai ∈ N(D(R)). Then (f)(D(R)) ⊆ N(D(R)), and
(f)(R) ⊆ N(D(R)). Let ((f)(R))k = 0, k > 0. Then equating leading
term to zero, we get (xmamR)k = 0. This implies on simplification that
xkm(amR)k = 0. Therefore (amR)k = 0 ⊆ P , for all P ∈ MinSpec(R).
So we have amR ⊆ P , for all P ∈ MinSpec(R). Therefore am ∈ P (R) =
N(R). Now xmam ∈ D(N(R)) ⊆ N(D(R)) implies that
∑
m−1
i=0 xiai ∈
N(D(R)), and with the same process, in a finite number of steps, it can
be seen that ai ∈ P (R) = N(R), 0 ≤ i ≤ m−1. Therefore f ∈ D(N(R)).
Hence N(D(R)) ⊆ D(N(R)) and the result.
Theorem 1.1. Let R be a Noetherian Q-algebra and δ be a derivation
of R. Then P ∈ MinSpec(D(R)) if and only if P = D(P ∩ R) and
P ∩ R ∈ MinSpec(R).
Proof. Let P1 ∈ MinSpec(R). Then δ(P1) ⊆ P1 by Proposition 1.1.
Therefore by [11, 14.2.5 (ii)], D(P1) ∈ Spec(D(R)). Suppose P2 ⊂
D(P1) is a minimal prime ideal of D(R). Then
P2 = D(P2 ∩ R) ⊂ D(P1) ∈ MinSpec(D(R)).
So P2 ∩ R ⊂ P1 which is not possible.
Conversely suppose that P ∈ MinSpec(D(R)). Then P∩R ∈ Spec(R)
by Lemma 2.21 of Goodearl and Warfield [6]. Let P1 ⊂ P ∩ R be a
minimal prime ideal of R. Then D(P1) ⊂ D(P ∩ R) and as in first
paragraph D(P1) ∈ Spec(D(R)), which is a contradiction. Hence P∩R ∈
MinSpec(R).
We are now in a position to prove the main result of this section in
the form of the following Theorem.
Theorem 1.2. Let R be a 2-primal Noetherian Q-algebra. Then D(R)
is 2-primal Noetherian.
Proof. R is Noetherian implies D(R) is Noetherian follows from Hilbert
Basis Theorem, namely Theorem 1.12 of Goodearl and Warfield [6]. Now
R is 2-primal implies N(R) = P (R) and Proposition (1.1) implies that
δ(N(R)) ⊆ N(R). Therefore D(N(R)) = D(P (R)). Now by Proposi-
tion 1.2 D(N(R)) = N(D(R)).
V. K. Bhat 157
We now show that D(P (R)) = P (D(R)). It is easy to see that
D(P (R)) ⊆ P (D(R)).
Now let
g =
t
∑
i=0
xibi ∈ P (D(R)).
Then g ∈ Pi, for all Pi ∈ MinSpec(D(R)). Now Theorem 1.1 implies that
there exists Ui ∈ MinSpec(R) such that Pi = D(Ui). Now it can be seen
that Pi are distinct implies that Ui are distinct. Therefore g ∈ D(Ui).
This implies that bi ∈ Ui. Thus we have bi ∈ Ui, for all Ui ∈ MinSpec(R).
Therefore bi ∈ P (R), which implies that g ∈ D(P (R)). So we have
P (D(R)) ⊆ D(P (R)), and hence D(P (R)) = P (D(R)).
Thus we have
P (D(R)) = D(P (R)) = D(N(R)) = N(D(R)).
Hence D(R) is 2-primal.
Question 1.1. Let R be a 2-primal Noetherian Q-algebra. Is O(R) 2-
primal (even if σ(δ(a)) = δ(σ(a)), for all a ∈ R)?
The main difficulty is that Proposition 1.2 and Theorem 1.1 do not
hold.
A step towards the answer of the above question is the following
Proposition and may give some idea:
Proposition 1.3. Let R be a ring. Let σ be an automorphism of R and
δ be a σ-derivation od R. Then:
1. For any completely prime ideal P of R with δ(P ) ⊆ P , P [x, σ, δ] is
a completely prime ideal of R[x, σ, δ].
2. For any completely prime ideal U of R[x, σ, δ], U∩R is a completely
prime ideal of R.
Proof. See [4, Proposition 2.5].
References
[1] S. Annin, Associated primes over skew polynomial rings // Comm. Algebra 30
(2002), 2511–2528.
[2] N. Argac and N. J. Groenewald, A generalization of 2-primal near rings // Ques-
tiones Mathematicae, 27 (2004), N 4, 397–413.
[3] V. K. Bhat, A note on Krull dimension of skew polynomial rings // Lobachevskii
J. Math, 22 (2006), 3–6.
158 Differential operator...
[4] V. K. Bhat, On 2-primal Ore extensions // Ukrainian Math. Bull., 4 (2007), N 2,
173–179.
[5] P. Gabriel, Representations des Algebres de Lie Resoulubles D Apres J. Dixmier.
In Seminaire Bourbaki, 1968–69, Pp. 1–22, Lecture Notes in Math. No 179, Berlin,
Springer-Verlag, 1971.
[6] K. R. Goodearl and R. B. Warfield Jr, An introduction to non-commutative
Noetherian rings, Cambridge Uni. Press, 1989.
[7] C. Y. Hong and T. K. Kwak, On minimal strongly prime ideals // Comm. Algebra
28(10) (2000), 4868–4878.
[8] 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.
[9] N. K. Kim and T. K. Kwak, Minimal prime ideals in 2-primal rings // Math.
Japonica 50(3) (1999), 415–420.
[10] G. Marks, On 2-primal Ore extensions // Comm. Algebra, 29 (2001), N 5, 2113–
2123.
[11] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley
1987; revised edition: American Math. Society, 2001.
Contact information
Vijay K. Bhat School of Applied Physics and Mathematics,
SMVD University, P/o Kakryal,
Udhampur, J and K,
India–182121
E-Mail: vijaykumarbhat2000@yahoo.com
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