Prime radical of Ore extensions over δ -rigid rings
Let R be a ring. Let σ be an automorphism of R and δ be a σ-derivation of R. We say that R is a δ-rigid ring if aδ(a)∈P(R) implies a∈P(R), a∈R; where P(R) is the prime radical of R. In this article, we find a relation between the prime radical of a δ-rigid ring R and that of R[x,σ,δ]. We generalize...
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Інститут прикладної математики і механіки НАН України
2009
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| Назва видання: | Algebra and Discrete Mathematics |
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| Цитувати: | Prime radical of Ore extensions over δ -rigid rings / V.K. Bhat // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 14–19. — Бібліогр.: 13 назв. — англ. |
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irk-123456789-1533792019-06-15T01:26:19Z Prime radical of Ore extensions over δ -rigid rings Bhat, V.K. Let R be a ring. Let σ be an automorphism of R and δ be a σ-derivation of R. We say that R is a δ-rigid ring if aδ(a)∈P(R) implies a∈P(R), a∈R; where P(R) is the prime radical of R. In this article, we find a relation between the prime radical of a δ-rigid ring R and that of R[x,σ,δ]. We generalize the result for a Noetherian Q-algebra (Q is the field of rational numbers). 2009 Article Prime radical of Ore extensions over δ -rigid rings / V.K. Bhat // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 14–19. — Бібліогр.: 13 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 16-XX; 16P40,16P50,16U20. http://dspace.nbuv.gov.ua/handle/123456789/153379 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let R be a ring. Let σ be an automorphism of R and δ be a σ-derivation of R. We say that R is a δ-rigid ring if aδ(a)∈P(R) implies a∈P(R), a∈R; where P(R) is the prime radical of R. In this article, we find a relation between the prime radical of a δ-rigid ring R and that of R[x,σ,δ]. We generalize the result for a Noetherian Q-algebra (Q is the field of rational numbers). |
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Article |
| author |
Bhat, V.K. |
| spellingShingle |
Bhat, V.K. Prime radical of Ore extensions over δ -rigid rings Algebra and Discrete Mathematics |
| author_facet |
Bhat, V.K. |
| author_sort |
Bhat, V.K. |
| title |
Prime radical of Ore extensions over δ -rigid rings |
| title_short |
Prime radical of Ore extensions over δ -rigid rings |
| title_full |
Prime radical of Ore extensions over δ -rigid rings |
| title_fullStr |
Prime radical of Ore extensions over δ -rigid rings |
| title_full_unstemmed |
Prime radical of Ore extensions over δ -rigid rings |
| title_sort |
prime radical of ore extensions over δ -rigid rings |
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Інститут прикладної математики і механіки НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/153379 |
| citation_txt |
Prime radical of Ore extensions over δ
-rigid rings / V.K. Bhat // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 14–19. — Бібліогр.: 13 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT bhatvk primeradicaloforeextensionsoverdrigidrings |
| first_indexed |
2025-07-14T04:36:17Z |
| last_indexed |
2025-07-14T04:36:17Z |
| _version_ |
1837595652703387648 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 1. (2009). pp. 14 – 19
c© Journal “Algebra and Discrete Mathematics”
Prime radical of Ore extensions over δ-rigid rings
V. K. Bhat
Communicated by M. Ya. Komarnytskyj
Abstract. Let R be a ring. Let σ be an automorphism of
R and δ be a σ-derivation of R. We say that R is a δ-rigid ring if
aδ(a) ∈ P (R) implies a ∈ P (R), a ∈ R; where P(R) is the prime
radical of R. In this article, we find a relation between the prime
radical of a δ-rigid ring R and that of R[x, σ, δ]. We generalize
the result for a Noetherian Q-algebra (Q is the field of rational
numbers).
1. Introduction
A ring R always means an associative ring. Q denotes the field of rational
numbers, and Z denotes the ring of integers unless other wise stated.
Spec(R) denotes the set of all prime ideals of R. Min.Spec(R) denotes
the sets of all minimal prime ideals of R. P(R) and N(R) denote the
prime radical and the set of all nilpotent elements of R respectively.
Let R be a ring. Let σ be an automorphism and δ be a σ-derivation of
R. Recall that R[x, σ, δ] is the usual polynomial ring with coefficients in
R and we consider any f(x) ∈ R[x, σ, δ] to be of the form f(x) =
∑
xiai,
0 ≤ i ≤ n. Multiplication in R[x, σ, δ] is subject to the relation ax =
xσ(a) + δ(a) for a ∈ R.
Ore-extensions including skew-polynomial rings and differential oper-
ator rings have been of interest to many authors. See [1, 2, 4, 5, 7, 11, 12].
In [1] associated prime ideals of skew polynomial rings have been dis-
cussed. In [5] it is shown that if R is embeddable in a right Artinian
ring and if characteristic of R is zero, then the differential operator ring
2000 Mathematics Subject Classification: 16-XX; 16P40,16P50,16U20.
Key words and phrases: Radical, automorphism, derivation, completely prime,
δ-ring, Q-algebra.
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.V. K. Bhat 15
R[x, δ] embeds in a right Artinian ring, where δ is a derivation of R. It
is also shown in [5] that if R is a commutative Noetherian ring and σ is
an automorphism of R, then the skew-polynomial ring R[x, σ] embeds in
an Artinian ring. In [2] it is proved that if R is a ring which is an order
in an Artinian ring, then R[x, σ, δ] is also an order in an Artinian ring.
Some authors have worked on R[x, σ, δ] when R is 2-primal. Recall
that a ring R is 2-primal if N(R) = P(R). R is 2-primal if and only if P(R)
is completely semiprime (i.e. a2 ∈ P (R) implies a ∈ P (R), a ∈ R). We
note that any reduced ring is 2-primal, and any commutative ring is also
2-primal. The nature of nil radical, prime ideals, minimal prime ideals,
prime radical of R[x, σ, δ] has been investigated, and relations between R
and R[x, σ, δ] have been obtained in some cases. For further details on
2-primal rings, we refer the reader to [4, 6, 8, 10, 13].
Recall that in [11], a ring R is called σ-rigid if aσ(a) = 0 implies that
a = 0 for a ∈ R. In [12], a ring R is called a σ(∗)-ring if aσ(a) ∈ P (R)
implies a ∈ P (R) and a relation has been established between a σ(∗)-ring
and a 2-primal ring. The property is also extended to R[x, σ].
Motivated by these developments, in this article, we define a δ-rigid
ring (Definition (2.1)), and establish a relation between a δ-rigid ring and
a 2-primal ring. We also find a relation between the prime radical of a δ-
rigid ring R and that of R[x, σ, δ]. We also discuss completely prime ideals
and the prime radical of a 2-primal ring R and try to relate completely
prime ideals of a ring R with the completely prime ideals of R[x, σ, δ].
This is given in Proposition (2.4). We also find a relation between the
prime radical of a 2-primal ring R and that of R[x, σ, δ]. This is given in
Theorem (2.6). We generalize this result for a Noetherian Q-algebra R.
This is given in Corollary (2.8).
2. Main Result
Let R be a ring. Let σ be an automorphism of R and δ be a σ-derivation
of a ring R. Recall that an ideal I of a ring R is called σ-invariant if σ(I)
= I and is called δ-invariant if δ(I) ⊆ I. Also I is called completely prime
if ab ∈ I implies a ∈ I or b ∈ I for a, b ∈ R. With this we have the
following definition:
Definition 2.1. Let R be a ring. Let σ be an automorphism of R and
δ be a σ-derivation of R. We say that R is a δ-rigid ring if aδ(a) ∈ P (R)
implies a ∈ P (R), a ∈ R. We note that a ring R with identity 1 is not a
δ-rigid ring as 1δ(1) = 0
We note that all σ-derivations need not satisfy the property (aδ(a) ∈
P (R) implies a ∈ P (R), a ∈ R). For example the following:
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.16 Prime radical of Ore extensions over δ-rigid rings
Consider R = (aij)2,2, the set of all 2x2 matrices over the ring nZ,
n > 1 with a21 = 0. Define σ: R → R by σ(aij) = (bij), where bij = aij
except that b12 = −a12 . Then it can be seen that δ is an automorphism
of R. Now define δ : R → R by δ(aij) = (cij), where cij = 0 except that
c12 = 2a12 + a22 − a11. Then it can be seen that δ is a σ-derivation of R.
But R is not a δ-rigid ring, as for A = (aij)2,2, with aij =0 except a22 =
1, Aδ(A) = (0).
Proposition 2.2. Let R be a 2-primal ring. Let σ be an automorphism
of R and δ be a σ-derivation of R such that δ(P (R)) ⊆ P (R). Let P ∈
Min.Spec(R) be such that σ(P ) = P. Then δ(P ) ⊆ P .
Proof. The proof follows from Theorem (3.6) and Lemma (3.2) of [9]. We
give a sketch of the proof.
Let P ∈ Min.Spec(R) with σ(P ) = P. Let a ∈ P . Then there exists
b /∈ P such that ab ∈ P (R) by Corollary (1.10) of [11]. Now we have
δ(P (R)) ⊆ P (R). Therefore δ(ab) = δ(a)σ(b) + aσ(b) ∈ P (R) ⊆ P . So
we have δ(a)σ(b) ∈ P . But σ(b) /∈ P , and therefore δ(a) ∈ P as by
Proposition (1.11) of [12] P is completely prime. Hence δ(P ) ⊆ P .
We now give a relation between a δ-rigid ring and a 2-primal ring.
Theorem 2.3. Let R be a δ-rigid ring. Let σ be an automorphism of
R such that σ(P (R)) = P (R), and δ be a σ-derivation of R such that
δ(P (R)) ⊆ P (R). Then R is 2-primal.
Proof. Define a map ∂ : R/P (R) → R/P (R) by ∂(a + P (R)) = δ(a) +
P (R) for a ∈ R and τ : R/P (R) → R/P (R) a map by τ(a + P (R))
= σ(a) + P (R) for a ∈ R. Now it is easy to see that that τ is an
automorphism of R/P (R). Also for any a + P(R), b + P (R) ∈ R/P (R);
∂((a + P (R))(b + P (R)) = ∂(ab + P (R)) = δ(ab) + P (R) = δ(a)σ(b) +
aδ(b) + P (R) = (δ(a) + P (R))(σ(b) + P (R)) + (a + P (R))(δ(b) + P (R))
= ∂(a + P (R))τ(b + P (R)) + (a + P (R))∂(b + P (R)), and it is obvious
that ∂(a + P (R) + b + P (R)) = ∂(a + P (R)) + ∂(b + P (R)). Therefore
∂ is a τ − derivation of R/P(R). Now a δ(a) ∈ P (R) if and only if
(a + P (R))∂(a + P (R)) = P(R) in R/P(R). Thus, as in Proposition (5)
of [7], R is a reduced ring and hence R is 2-primal.
We notice that a 2-primal ring need not be a δ-rigid ring, as can be
seen from the following example.
Consider R = Z2 ⊕ Z2. Then R is a commutative reduced ring, and
so is a 2-primal ring. Define a map σ : R → R by σ(a, b) = (b, a). Then
σ is an automorphism of R. Now define a map δ : R → R by δ(a, b) =
(a-b, 0). Then δ is a σ-derivation of R. But R is not a δ-rigid ring, as (0,
1)δ(0, 1) = (0, 0).
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.V. K. Bhat 17
Proposition 2.4. Let R be a ring. Let σ be an automorphism of R and
δ be a σ-derivation of R. Then:
1. For any completely prime ideal P of R with δ(P ) ⊆ P and σ(P ) =
P , P [x, σ, δ] is a completely prime ideal of R[x, σ, δ].
2. For any completely prime ideal Q of R[x, σ, δ], Q∩R is a completely
prime ideal of R.
Proof. See Proposition (2.5) of [3].
The above discussion leads to the following question:
Is δ(Q∩R) ⊆ Q∩R in Proposition (2.4)? If so, is Q = (Q∩R)[x, σ, δ]?
The question remains to be answered, but in this connection we note that
σ and δ can be extended to R[x, σ, δ] by taking σ(x) = x and δ(x) = 0.
In other words, σ(xa) = xσ(a) and δ(xa) = xδ(a) for all a ∈ R.
Corollary 2.5. Let R be a δ-rigid ring. Let σ be an automorphism of
R and δ be a σ-derivation of R such that δ(P (R)) ⊆ P (R). Let P ∈
Min.Spec(R) be such that σ(P ) = P. Then P [x, σ, δ] is a completely
prime ideal of R[x, σ, δ].
Proof. R is 2-primal by Theorem (2.3), and so by Proposition (2.2)
δ(P ) ⊆ P . Further more P is a completely prime ideal of R by Proposition
(1.11) of [12]. Now use Proposition (2.4).
Theorem 2.6. Let R be a δ-rigid ring. Let σ be an automorphism of
R and δ be a σ-derivation of R such that δ(P (R)) ⊆ P (R) and σ(P ) =
P for all P ∈ Min.Spec(R). Then R[x, σ, δ] is 2-primal if and only if
P (R)[x, σ, δ] = P (R[x, σ, δ]).
Proof. Let R[x, σ, δ] be 2-primal. Let P ∈ Min.Spec(R). By Corollary
(2.5) P [x, σ, δ] is a completely prime ideal of R[x, σ, δ], and therefore
P (R[x, σ, δ]) ⊆ P (R)R[x, σ, δ]. One may see Proposition (3.8) of [9] also.
Let f(x) =
∑
xjaj ∈ P (R)[x, σ, δ], 0 ≤ i ≤ n. Now R is a 2-primal sub
ring of R[x, σ, δ] by Theorem (2.3). This implies that aj is nilpotent and
thus aj ∈ N(R[x, σ, δ]) = P (R[x, σ, δ], and so we have xjaj ∈ P (R[x, σ, δ])
for each j. Therefore f(x) ∈ P (R[x, σ, δ]). Hence we have P (R)[x, σ, δ]
= P (R[x, σ, δ]).
Conversely suppose P (R)[x, σ, δ] = P (R[x, σ, δ]). We will show that
R[x, σ, δ] is 2-primal. Let g(x) =
∑
xibi ∈ R[x, σ, δ], 0 ≤ i ≤ n be such
that (g(x))2 ∈ P (R[x, σ, δ]) = P (R)[x, σ, δ]. Then by an easy induction
and by using the fact that P(R) is completely semiprime by Theorem
(2.3), it can be easily seen that bi ∈ P (R) for all bi, 0 ≤ i ≤ n. This
means that f(x) ∈ P (R)[x, σ, δ] = P (R[x, σ, δ]). Therefore P (R[x, σ, δ]
is completely semiprime. Hence R[x, σ, δ] is 2-primal.
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.18 Prime radical of Ore extensions over δ-rigid rings
We now generalize the above result for a Noetherian Q-algebra R, and
towards this we have the following:
Proposition 2.7. Let R be a Noetherian Q-algebra. Let σ be an auto-
morphism of R and δ be a σ-derivation of R such that σ(δ(a)) = δ(σ(a)),
for a ∈ R. Then:
1. σ(N(R)) = N(R)
2. If P ∈ Min.Spec(R) such that σ(P ) = P , then δ(P ) ⊆ P .
Proof. (1) Denote N(R) by N. We have σ(N) ⊆ N as σ(N) is a nilpotent
ideal of R. Now for any n ∈ N , there exists a ∈ R such that n = σ(a).
So I= σ−1(N)={a ∈ R such that σ(a) = n ∈ N} is an ideal of R. Now
I is nilpotent, therefore I ⊆ N , which implies that N ⊆ σ(N). Hence
σ(N) = N .
(2) Let T = {a ∈ P such that δk(a) ∈ P for all integers k ≥ 1}. Then
T is a δ-invariant ideal of R. Now it can be seen that T ∈ Spec(R), and
since P ∈ Min.Spec(R), we have T = P. Hence δ(P ) ⊆ P .
Corollary 2.8. Let R be a δ-rigid Noetherian Q-algebra. Let σ be an
automorphism of R and δ be a σ-derivation of R such that σ(δ(a)) =
δ(σ(a)), for a ∈ R. Let σ(P ) = P for all P ∈ Min.Spec(R). Then
R[x, σ, δ] is 2-primal if and only if P (R)[x, σ, δ] = P (R[x, σ, δ]).
Proof. Use Theorems (2.6) and (2.7).
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Contact information
V. K. Bhat School of Applied Physics and Mathematics,
SMVD University, P/o Kakryal, Katra, J
and K, India-182301
E-Mail: vijaykumarbhat2000@yahoo.com
Received by the editors: 14.09.2007
and in final form 01.05.2009.
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