On separable and H-separable polynomials in skew polynomial rings of several variables
Let B be a ring with 1, and {ρ1,⋯,ρe} a set of automorphisms of B. Let B[X1,⋯,Xe;ρ1,⋯,ρe;{uij}] be the skew polynomial ring of automorphism type. In this paper, we shall give equivalent conditions that the residue ring of B[X1,⋯,Xe;ρ1,⋯,ρe;{uij}] by the ideal generated by a set {Xm11−u1,⋯,Xmee−ue} t...
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| Цитувати: | On separable and H-separable polynomials in skew polynomial rings of several variables / S.Ikehata // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 87–85. — Бібліогр.: 13 назв. — англ. |
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irk-123456789-1548702019-06-18T01:30:32Z On separable and H-separable polynomials in skew polynomial rings of several variables Ikehata, S. Let B be a ring with 1, and {ρ1,⋯,ρe} a set of automorphisms of B. Let B[X1,⋯,Xe;ρ1,⋯,ρe;{uij}] be the skew polynomial ring of automorphism type. In this paper, we shall give equivalent conditions that the residue ring of B[X1,⋯,Xe;ρ1,⋯,ρe;{uij}] by the ideal generated by a set {Xm11−u1,⋯,Xmee−ue} to be separable or H-separable over B. 2010 Article On separable and H-separable polynomials in skew polynomial rings of several variables / S.Ikehata // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 87–85. — Бібліогр.: 13 назв. — англ. 2000 Mathematics Subject Classification:16S30, 16W20 http://dspace.nbuv.gov.ua/handle/123456789/154870 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let B be a ring with 1, and {ρ1,⋯,ρe} a set of automorphisms of B. Let B[X1,⋯,Xe;ρ1,⋯,ρe;{uij}] be the skew polynomial ring of automorphism type. In this paper, we shall give equivalent conditions that the residue ring of B[X1,⋯,Xe;ρ1,⋯,ρe;{uij}] by the ideal generated by a set {Xm11−u1,⋯,Xmee−ue} to be separable or H-separable over B. |
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Article |
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Ikehata, S. |
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Ikehata, S. On separable and H-separable polynomials in skew polynomial rings of several variables Algebra and Discrete Mathematics |
| author_facet |
Ikehata, S. |
| author_sort |
Ikehata, S. |
| title |
On separable and H-separable polynomials in skew polynomial rings of several variables |
| title_short |
On separable and H-separable polynomials in skew polynomial rings of several variables |
| title_full |
On separable and H-separable polynomials in skew polynomial rings of several variables |
| title_fullStr |
On separable and H-separable polynomials in skew polynomial rings of several variables |
| title_full_unstemmed |
On separable and H-separable polynomials in skew polynomial rings of several variables |
| title_sort |
on separable and h-separable polynomials in skew polynomial rings of several variables |
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Інститут прикладної математики і механіки НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/154870 |
| citation_txt |
On separable and H-separable polynomials in skew polynomial rings of several variables / S.Ikehata // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 87–85. — Бібліогр.: 13 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT ikehatas onseparableandhseparablepolynomialsinskewpolynomialringsofseveralvariables |
| first_indexed |
2025-07-14T06:56:17Z |
| last_indexed |
2025-07-14T06:56:17Z |
| _version_ |
1837604461013368832 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 10 (2010). Number 2. pp. 87 – 95
c© Journal “Algebra and Discrete Mathematics”
On separable and H-separable polynomials in
skew polynomial rings of several variables
Shûichi Ikehata
Communicated by V. V. Kirichenko
Abstract. Let B be a ring with 1, and {ρ1, · · · , ρe} a
set of automorphisms of B. Let B[X1, · · · , Xe; ρ1, · · · , ρe; {uij}]
be the skew polynomial ring of automorphism type. In this pa-
per, we shall give equivalent conditions that the residue ring of
B[X1, · · · , Xe; ρ1, · · · , ρe; {uij}] by the ideal generated by a set
{Xm1
1
− u1, · · · , X
me
e − ue} to be separable or H-separable over
B.
1. Introduction
In [4], K. Hirata and K. Sugano generalized the notion of separable algebras
to that of separable extensions of a ring. A ring extension T/S is called a
separable extension if the T -T -homomorphism of T ⊗S T onto T defined
by a⊗ b → ab splits, and T/S is called an H-separable extension if T ⊗S T
is T -T -isomorphic to a direct summand of a finite direct sum of copies of
T . As is well known an H-separable extension is a separable extension.
Throughout this paper, B will mean a ring with identity 1, ρ an auto-
morphism of B, and Z the center of B. Let B[X; ρ] be the skew polynomial
ring in which the multiplication is given by bX = Xρ(b) (b ∈ B). A monic
polynomial f in B[X; ρ] such that fB[X; ρ] = B[X; ρ]f is called a sepa-
rable (resp. H-separable) polynomial if the residue ring B[X; ρ]/fB[X; ρ]
is a separable (resp. H-separable) extension of B. Separable polynomials
in skew polynomial rings are extensively studied by Kishimoto, Naga-
hara, Miyashita, Szeto, Xue and the author (see References). In [9, 10],
2000 Mathematics Subject Classification: 16S30, 16W20.
Key words and phrases: H-separable polynomial, separable extension, skew
polynomial ring.
88 On separable and H -separable polynomials
Kishimoto studied some special type of separable polynomials in skew poly-
nomial rings. In [12], Nagahara gave a thorough investigation of separable
polynomials of degree 2. Miyashita [11] studied systematically separable
polynomials and Frobenius polynomials. The following is a theorem of Y.
Miyashita which characterizes the sparability of Xn − u in B[X; ρ].
Proposition 1.1 ([11, Theorem 3.1]). Let f = Xn − u be in B[X; ρ].
Then the following conditions are equivalent:
(1) f is a separable polynomial in B[X; ρ].
(2) (i) ρ(u) = u, and αu = uρn(α) for all α ∈ B,
(ii) u is invertible in Bρ, and there exists an element z ∈ Z such
that
z + ρ(z) + · · ·+ ρn−1(z) = 1.
In [6, 7, 8], the author has studied H-separable polynomials in skew
polynomial rings. If the coefficient ring is commutative, the existence of
an H-separable polynomial in a skew polynomial ring has been charac-
terized in terms of Azumaya algebras and Galois extensions. Recall that
a ring extension T/S is called G-Galois, if there exist a finite group G
of automorphisms of T such that S = TG (the fix ring of G in T ) and∑
i xiσ(yi) = δ1,σ (σ ∈ G) for some finite number of elements xi, yi ∈ T .
In [8], the author proved that B[X; ρ] contains an H-separable polynomial
of prime degree if and only if the center Z of B is a Galois extension over
Zρ. In [13], G. Szeto and L. Xue have succeeded in a general degree case.
Proposition 1.2 ([13, Theorem 3.6]). Let f = Xn − u be in B[X; ρ].
Then the following conditions are equivalent:
(1) f is an H-separable polynomial in B[X; ρ].
(2) (i) ρ(u) = u, and αu = uρn(α) for all α ∈ B,
(ii) u is invertible in Bρ, and Z/Zρ is a G-Galois extension, where
G is the group generated by ρ|Z of degree n.
The purpose of this paper is to generalize these results to the skew
polynomial rings in several variables.
2. Preliminaries
First of all, we shall state some elementary properties of separable and
H-separable extensions which are useful in our subsequent study.
S. Ikehata 89
Lemma 2.1 ([4, Proposition 2.5 (1)]). Let R ⊃ S ⊃ T be ring extensions.
If R/S and S/T are separable (resp. H-separable) extensions, then R/T
is also a separable (resp. H-separable) extension.
Lemma 2.2 ([4, Proposition 2.5 (2)]). Let R ⊃ S ⊃ T be ring extensions.
If R/T is a separable extension, then R/S is a separable extension.
Lemma 2.3 ([3, Proposition 4.3]). Let R ⊃ S ⊃ T be ring extensions. If
R/T is an H-separable extension and S/T is a separable extension, then
R/S is an H-separable extension.
The following lemma must be well known but we could not find in the
literature, so we give a proof.
Lemma 2.4. Let Z be a commutative ring, and G = N×K a finite abelian
group of automorphisms of Z. If Z/ZG is a G-Galois extension, Then
Z/ZN is an N -Galois extension and ZN/ZG is a K-Galois extension.
Proof. Since Z/ZG is a G-Galois extension, there exist a G-Galois coordi-
nate system {xi, yi} ⊂ Z such that
∑
i
xiσ(yi) = δ1,σ (σ ∈ G).
Then obviously, Z/ZN is an N -Galois extension. By [2, Lemma 1.6], there
exists an element c ∈ Z such that trN (c) =
∑
σ∈N σ(c) = 1. Then we can
easily see that ∑
i
xiτ(trN (yi)) = δ1,τ (τ ∈ K).
So we have ∑
i
trN (xic)τ(trN (yi)) = δ1,τ (τ ∈ K).
This means {trN (xic), trN (yi)} is a K-Galois coordinate system for
ZN/ZG.
3. Main results
We need some notations as given by K. Kishimoto [9], S. Ikehata [7] and
S. A. Amitsur and D. Saltman [1].
Let ρi (1 ≦ i ≦ e) be automorphisms of a ring B, and let uij (1 ≦
i, j ≦ e) be invertible elements in B such that
(i) uij = u−1
ji , and uii = 1,
(ii) ρiρjρ
−1
i ρ−1
j = (uij)ℓ(u
−1
ij )r,
(iii) uijρj(uik)ujk = ρi(ujk)uikρk(uij).
90 On separable and H -separable polynomials
Then the set of all polynomials in e indeterminates {X1, X2, · · · , Xe}
is
{
∑
Xν1
1 Xν2
2 · · ·Xνe
e bν1ν2···νe | bν1ν2···νe ∈ B, νk ≧ 0}
which is an associative ring such that the multiplication is defined by
aXi = Xiρi(a) (a ∈ B) and XiXj = XjXiuij (1 ≦ i, j ≦ e).
This ring is denoted by Re = B[X1, X2, · · · , Xe; ρ1, ρ2, · · · , ρe; {uij}]
and is called a skew polynomial ring of automorphism type.
Moreover, by Rk (0 ≦ k ≦ e), we denote the skew polynomial ring
B[X1, X2, · · · , Xk; ρ1, ρ2, · · · , ρk; {uij}] which is a subring of Re, where
R0 = B.
Remark 3.1. For a permutation π of {1, 2, · · · , k} (k ≦ e), we have a
B-ring automorphism
Rk
∼= B[Xπ(1), Xπ(2), · · · , Xπ(k); ρπ(1), ρπ(2), · · · , ρπ(k); {uπ(i)π(j)}]
which maps Xi to Xπ(i) (1 ≦ i ≦ k).
Now, assume further that there exist elements ui (1 ≦ i ≦ e) in B
such that
(iv) bui = uiρ
mi
i (b) (b ∈ B)
and
(v) ρj(ui)ujiρi(uji) · · · ρ
mi−1
i (uji) = ui (1 ≦ i ≦ e).
Then we have,
a(Xmi
i − ui) = (Xmi
i − ui)ρ
mi
i (a) (a ∈ B)
and
Xj(X
mi
i − ui) = (Xmi
i − ui)Xjujiρi(uji) · · · ρ
mi−1
i (uji) (1 ≦ i, j ≦ e).
This means that (Xmi
i − ui)Rk = Rk(X
mi
i − ui) is a two-sided ideal
of Rk for i ≦ k ≦ e. The mapping ρ̄i : Re → Re defined by
ρ̄i(
∑
Xν1
1 Xν2
2 · · ·Xνe
e bν1ν2···νe) =
=
∑
(X1u1i)
ν1(X2u2i)
ν2 · · · (Xeuei)
νeρi(bν1ν2···νe)
is an automorphism of Re which is an extension of ρi.
For each i (1 ≦ i ≦ e), we put here
Bi = B[X1, · · · , Xi−1, Xi+1, · · · , Xe; ρ1, · · · , ρi−1, ρi+1, · · · , ρe; {uij}].
S. Ikehata 91
Naturally, we have
Re = Bi[Xi; ρ̄i],
and
β(Xmi
i − ui) = (Xmi
i − ui)ρ̄
mi
i (β) (β ∈ Bi) and ρ̄i(ui) = ui,
where ρ̄i means ρ̄i|Bi.
Let M = (Xm1
1 − u1, X
m2
2 − u2, · · · , X
me
e − ue) be the two sided ideal
of Re generated by {Xm1
1 −u1, X
m2
2 −u2, · · · , X
me
e −ue}. Then the residue
ring Re/M is a free ring extension over B with a basis
{xν11 xν22 · · ·xνee | 0 ≦ νi < mi, 1 ≦ i ≦ e}, where xi = Xi +M ∈ Re/M .
Since ρ̄i(X
mj
j − uj) = (X
mj
j − uj)ρ
mj−1
j (uji)ρ
mj−2
j (uji) · · · ρj(uji)uji, we
obtain ρ̄i(M) = M . Hence, naturally ρ̄i induces the automorphism
ρ̄i : Re/M → Re/M , where we use the same notation ρ̄i.
Under the above notations, we shall prove our first theorem which is
a generalization of Proposition 1.1.
Theorem 3.2. The following are equivalent.
(1) Re/M is a separable extension of B.
(2) (i) ui ∈ U(Bρi) (1 ≦ i ≦ e).
(ii) There exists an element z ∈ Z such that
∑
0≦ν1<m1
∑
0≦ν2<m2
· · ·
∑
0≦νe<me
ρν11 ρν22 · · · ρνee (z) = 1.
(3) Xmi
i − ui is a separable polynomial in Bi[Xi; ρ̄i] for each i (1 ≦ i ≦
e).
(4) (i) ui ∈ U(Bρi) (1 ≦ i ≦ e).
(ii) There exist elements ci ∈ Zρ1,ρ2,··· ,ρi−1,ρi+1,··· ,ρe such that
ci + ρi(ci) + · · ·+ ρmi−1
i (ci) = 1.
Proof. (1) =⇒ (2). Let Mi be the ideal of Bi generated by {Xm1
1 −
u1, · · · , X
mi−1
i−1 −ui−1, X
mi+1
i+1 −ui+1, · · · , X
me
e −ue}. Since ρ̄i(Mi) = Mi,
we have that ρ̄i induces the automorphism ρ̄i : Bi/Mi → Bi/Mi. Then
we have
Re/M = (Bi/Mi)[Xi; ρ̄i]/(X
mi
i − ui)(Bi/Mi)[Xi; ρ̄i].
92 On separable and H -separable polynomials
Since Re/M ⊃ Bi/Mi ⊃ B and Re/M is a separable extension of B,
it follows from Lemma 2.2 that Re/M is also a separable extension of
Bi/Mi, that is, Xmi
i − ui is a separable polynomial in (Bi/Mi)[Xi; ρ̄i].
Then by Proposition 1.1, ui is invertible in B
ρ̄i
i , so is invertible in Bρi ,
and there exists an element yi in the center of Bi/Mi such that
yi + ρ̄i(yi) + · · ·+ ρ̄mi−1
i (yi) = 1.
Let ci be the constant term of yi. Then we see that ci is in Zρ1,ρ2,··· ,ρi−1,ρi+1,··· ,ρe
and ci + ρi(ci) + · · · + ρmi−1
i (ci) = 1. We put z = c1c2 · · · ce. Then it is
easy to see that
∑
0≦ν1<m1
∑
0≦ν2<m2
· · ·
∑
0≦νe<me
ρν11 ρν22 · · · ρνee (z) = 1.
This completes the proof of (1) =⇒ (2).
(2) =⇒ (3). We put here
ci =
∑
0≦ν1<m1
· · ·
∑
0≦νi−1<mi−1
∑
0≦νi+1<mi+1
· · ·
∑
0≦νe<me
ρν11 · · · ρ
νi−1
i−1 ρ
νi+1
i+1 · · · ρνee (z).
Then we obtain ci ∈ Zρ1,ρ2,··· ,ρi−1,ρi+1,··· ,ρe , and
ci + ρi(ci) + · · ·+ ρmi−1
i (ci) =
=
∑
0≦ν1<m1
∑
0≦ν2<m2
· · ·
∑
0≦νe<me
ρν11 ρν22 · · · ρνee (z) = 1.
Since ci is in the center of Bi, X
mi
i − ui is a separable polynomial in
Bi[Xi; ρ̄i] by Proposition 1.1.
(3) =⇒ (4). By Proposition 1.1, there exists yi in the center of Bi
such that yi + ρ̄i(yi) + · · ·+ ρ̄mi−1
i (yi) = 1. Considering the constant term
of yi, we have (4).
(4) =⇒ (1). We put here
S0 = B and S1 = B[X1; ρ1]/(X
m1
1 − u1)B[X1; ρ1],
and for each 1 ≦ i ≦ e,
Si = Si−1[Xi; ρ̄i]/(X
mi
i − ui)Si−1[Xi; ρ̄i],
where ρ̄i : Si−1 → Si−1 is a natural extension of ρi. Then, we have
Re/M = Se ⊃ Se−1 ⊃ · · · ⊃ S1 ⊃ S0 = B.
It is clear that each Xmi
i − ui is a separable polynomial in Si−1[Xi; ρ̄i].
That is, Si is a separable extension of Si−1. By the Lemma 2.1, we have
Re/M is a separable extension of B.
S. Ikehata 93
The following is a main theorem concerning to an H-separable exten-
sion which is a generalization of Proposition 1.2. We also use the notations
in the proof of the previous theorem.
Theorem 3.3. The following are equivalent.
(1) Re/M is an H-separable extension of B, and the centralizers of B
in Re/M , VRe/M (B) = Z.
(2) Xmi
i − ui is an H-separable polynomial in Si−1[Xi; ρ̄i] for each
i (1 ≦ i ≦ e).
(3) (i) ui ∈ U(Bρi) (1 ≦ i ≦ e).
(ii) The order of (ρi|Z) = mi (1 ≦ i ≦ e), the set {ρi|Z |1 ≦ i ≦ e}
generates an abelian group < ρ1|Z > × < ρ2|Z > × · · · × <
ρe|Z >= G, and Z/ZG is a G-Galois extension.
Proof. (3) =⇒ (2). We consider the following tower
Z ⊃ Zρ1 ⊃ Zρ1,ρ2 ⊃ · · · ⊃ Zρ1,··· ,ρe = ZG.
Since G =< ρ1|Z > × < ρ2|Z > × · · · × < ρe|Z > and Z/ZG is a G-
Galois extension of order m1m2 · · ·me, it follows from Lemma 2.4 that
Zρ1,··· ,ρi−1/Zρ1,··· ,ρi−1,ρi is a < ρi|Z
ρ1,··· ,ρi−1 >- Galois extension of order
mi for each i (1 ≦ i ≦ e). Then an easy induction shows that the center of
Si−1 is equal to Zρ1,··· ,ρi−1 . Thus, Xmi
i − ui is an H-separable polynomial
in Si−1[Xi; ρ̄i] by Proposition 1.2.
(2) =⇒ (1), (3). We consider the following tower
Re/M = Se ⊃ · · · ⊃ Si ⊃ Si−1 ⊃ · · · ⊃ S1 ⊃ S0 = B.
Since Xmi
i − ui is an H-separable polynomial in Si−1[Xi; ρ̄i],Si/Si−1 is
an H-separable extension. Hence by Lemma 2.1, Re/M is an H-separable
extension of B. To prove VRe/M (B) = Z, we shall show that the group
G generated by {ρ1|Z, ρ2|Z, · · · , ρe|Z} is a direct product < ρ1|Z > × <
ρ2|Z > × · · · × < ρe|Z >, and Z/ZG is a G-Galois extension. By an
induction, it is easy to verify that the center of Si−1 is equal to Zρ1,··· ,ρi−1 ,
and Zρ1,··· ,ρi−1/Zρ1,··· ,ρi−1,ρi is a < ρi|Z
ρ1,··· ,ρi−1 >- Galois extension of
order mi. Since (ρi|Z)
mi = mi (1 ≦ i ≦ e), G must be a direct product,
that is, G =< ρ1|Z > × < ρ2|Z > × · · · × < ρe|Z >, and Z/ZG is a G-
Galois extension. By a computation using a G-Galois coordinate system
for Z/ZG, we can easily see that VRe/M (B) = Z.
(1) =⇒ (2). Consider the following.
Re/M = (Bi/Mi)[Xi; ρ̄i]/(X
mi
i − ui)(Bi/Mi)[Xi; ρ̄i] ⊃ Bi/Mi ⊃ B.
94 On separable and H -separable polynomials
By the previous theorem, Bi/Mi is a separable extension of B. Then
by Lemma 2.3, we have Re/M is an H-separable extension of Bi/Mi,
that is, Xmi
i − ui is an H-separable polynomial in (Bi/Mi)[Xi; ρ̄i]. Since
VRe/M (B) = Z, the center of Bi/Mi is equal to Zρ1,ρ2,··· ,ρi−1,ρi+1,··· ,ρe .
Hence Zρ1,ρ2,··· ,ρi−1,ρi+1,··· ,ρe is a < ρi|Z
ρ1,ρ2,··· ,ρi−1,ρi+1,··· ,ρe >-Galois ex-
tension of Zρ1,ρ2,··· ,ρe . By using the same Galois coordinate system, we
see that Zρ1,ρ2,··· ,ρi−1 is a < ρi|Z
ρ1,ρ2,··· ,ρi−1,ρi >-Galois extension of
Zρ1,ρ2,··· ,ρi . Thus, Xmi
i − ui is an H-separable polynomial in Si−1[Xi; ρ̄i]
for each i (1 ≦ i ≦ e) by Proposition 1.2.
Remark 3.4. In case e = 1, the condition VRe/M (B) = Z in Theorem
3.3 (1) is superfluous.
We conclude our study with an example of non separable extension
R2/M of B, where M = (X2
1 − 1, X2
2 − 1) and R2 = B[X1, X2; ρ1, ρ2],
while each X2
i − 1 is a separable polynomial in B[X; ρi] (i = 1, 2).
Example 3.5. Let k be a field of a characteristic 2, B = k ⊕ k, and
ρ : B → B an automorphism defined by ρ(a, b) = (b, a). Let ρ1 = ρ2 =
ρ. Then we consider the skew polynomial ring B[X1, X2; ρ1, ρ2] such
that αX1 = X1ρ1(α), αX2 = X2ρ2(α) (α ∈ B), X1X2 = X2X1, that is,
u12 = u21 = 1. Since (1, 0) + ρ(1, 0) = (1, 1), each X2
i − 1 is a separable
polynomial in B[X; ρi] (i = 1, 2). We put R = B[X1, X2; ρ1, ρ2], and
M = the ideal generated by {X2
1 − 1, X2
2 − 1}. Then the residue ring
R/M is not a separable extension of B. Because for any (a, b) ∈ B,
(a, b) + ρ1(a, b) + ρ2(a, b) + ρ1ρ2(a, b) = 0.
Acknowledgement. This work was done while the author was visiting
at the Mathematics Department of Bradley University in spring 2009. He
expresses his gratitude to Professor George Szeto and Professor Larry
Xue for many useful discussions and the hospitality of the Mathematics
Department of Bradley University.
References
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Contact information
S. Ikehata Department of Environmental and Mathe-
matical Science, Faculty of Environmental
Science and Technology, Okayama University,
Tsushima, Okayama 700-8530, Japan
E-Mail: ikehata@ems.okayama-u.ac.jp
Received by the editors: 09.04.2009
and in final form 28.02.2011.
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