On Separable Extensions of Noncommutative Rings(Algebras, Languages, Computations and their Applications)

On Separable

Extensions of

Noncommutative

Rings

岡山大学・環境理工学部

,

池畑秀一

(Sh\^uichi IKEHATA)

Faculty

of Environmental

Science

and

Techmology

Okyama University

In [2], K. HirataandK. Suganogeneralized the notionofseparable algebrasdefining

separubleextensions of aring. Aring extension $T/S$ is caUed asepamble extension, if the T-T-homomorphism of $T\otimes_{S}T$ onto $T$ defined by $a\otimes barrow ab$ splits, and _$T/S$

is caUed

an

$H$-sepamble extension, if$T\otimes_{S}T$ is T-T-isomorphic to adirect summtd

of afinite direct

sum

of$\infty pies$ of T. As is well known every $H$-separable extension is

aseparable $e$xtension.

Throughout this paper, $BwiU$ mean _{aring with identity 1,} $\rho$ an automorphism

of $B$, and $Z$ the center of B. Let $B[X;\rho]$ be the skew polynomial ring in which the

multiplication is given by $bX=X\rho(b)(b\in B)$

.

Amonic polynomial $f$ in $B[X;\rho]$

with $fB[X;\rho]=B[X;\rho]f$ is cafed aseparable _(resp. $H$-separable)polynomial if the factor ring $B[X;\rho]/fB[X;\rho]$ is aseparable (resp. $H$-separable)extension of $B$

.

Separable polynomials in skew polynomial rings

are

extensively studied by

Kishi-moto, Nagahara, Miyashita, Szeto, Xue and the author (see Referenoes). In $[21, 22]$,

Kishimoto studied

some

special type of separable polynomials in skew polynomial

rings. In [27], Nagahara

gave

athorough investigation of separable polynomial of degree 2. Miyashita [26] studied $systemati_{Ca}g_{y}$separable polynomials and Robenius

polynomials. He give acharacterization of aseparable polynomial.

The following Is atheorem of Y. Miyashita which characterizes separabhty of

$X^{n}-u$ in $B[X;\rho]$

.

Proposition 1. ([26, Theorem 3.1]) Let$f=X^{n}-u$ be in$B[X;\rho]$

.

Then the following

conditions

are

equivalent:

(1) $f$ is a separable polynomial in $B[X;\rho]$

.

(2) (a) $\rho(u)=u$, and $\alpha u=u\rho^{n}(\alpha)$

for

all $\alpha\in B$,

(b) $u$ is invertible in $U(B^{p})$, and there $e$zists an element $z\in Z$ such that $z+\rho(z)+\cdots+\rho^{n-1}(z)=1$

.

Remark0.1. The$\infty ndition(2)(a)$ in the Proposition 1 is equivalent to $(X^{\mathfrak{n}}-u)B[X;\rho]=$

$B[X;\rho](X^{n}-u)$

.

In [9, 10, 11], the author has studied H-separable pokynomials inskew polynomial

rings. Ifthe coefficient ring is commutative, the existenceofH-separable polynomials in skew polynomial rings has been characterized in terms of Azumaya algebras and Galois extensions. In [10], the author proved that $B[X;\rho]$ contains

an

H-separable

polynomial of prime degree if and only if the center $Z$ of$B$ is a Galois extension over This is anabstract and the detaikwill be published ekewhere.

$Z^{\rho}$

.

In [19], G. Szeto and L. Xue has succeeded in general degree

case.

The following

is their theorem.

Proposition 2. ([19, Theorem3.6]) Let$f=X^{n}-u$ be in $B[X;\rho]$

.

Then the following

conditions are equivalent:

(1) $f$ is an H-separable polynomial in $B[X_{j}\rho]$

.

(2) (a) $\rho(u)=u$, and $\alpha u=u\rho^{n}(\alpha)$

for

all $\alpha\in B_{f}$

(b) $u$ is invertible in $U(B^{p})$, and $Z/Z^{\rho}$ is a G-Galois extension, where $G$ is

the group generated by $\rho|Z$

of

degree $n$

.

The

pourpose

of this paper is to generalize these results to the skew polynomial

rings in several variables. We needsome notations as in K. Kishimoto [21], S. Ikehata

[5] and S. A. Amitsur and D. Saltman [1].

Let $\rho_{i}(1\leqq i\leqq e)$ be automorphisms of a ring $B$, and let $u_{ij}(1\leqq i,j\leqq e)$ be

invertible elements in $B$ such that

(i) $v_{\dot{a}j}=u_{ji}^{-1}$, and _{$u_{ii}=1$},

(ii) $\rho_{i}\rho_{j}\rho_{i}^{-1}\rho_{j}^{-1}=(u_{ij})_{\ell}(u_{ij}^{-1}),$,

(lli) $u_{j}\rho_{J}(u:k)u_{J^{k}}=\rho_{i}(u_{jk})tAk\rho_{k}(u_{ij})$.

Then the set of all polynomials in $e$ indeterminates

$\{\sum X_{1^{1}}^{\nu}X_{2^{2}}^{\nu}\cdots X_{e}^{\nu_{*}}b_{\nu_{1}\nu_{2}\cdots\nu_{\epsilon}}|b_{\nu_{1}\eta\cdots\nu_{\epsilon}}\in B, \nu_{k}\geqq 0\}$

forms an associative ring ifwe definethe multiplication by the distributive law and the rules

$aX_{i}=X_{i}\rho_{i}(a)(a\in B)$ and $X_{i}X_{j}=X_{j}X_{i}u_{ij}(1\leqq i,j\leqq e)$

.

This ring is denoted by $R_{e}=B[X_{1}, X_{2}, \cdots X_{e};\rho_{1}, \rho_{2}, \cdots\rho_{e};\{u_{ij}\}]$ and is called

a skew polynomial ring of automorphism type. Moreover, by $R_{k}(0\leqq k\leqq e)$,

we

denote the skew polynomial ring $B[X_{1}, X_{2}, \cdots X_{k};\rho_{1)}\rho_{2)}\cdots ,\rho_{k};\{u_{1j}\}]$ which is

a

subring of$R_{e}$, where $R_{0}=B$

.

Remark0.2. For

a

permutation$\pi$ of$\{$1, 2,$\cdots$ ,$k\}(k\leqq e)$

,

wehave a B-ring

automor-phism $R_{k}\cong B[X_{\pi(1)}, X_{\pi(2)}, \cdots , X_{\pi(k)};\rho_{\pi(1)}, \rho_{\pi(2))}\cdots , \rho_{\pi(k)};\{u_{\pi(i)\pi(j)}\}]$ which maps

$X_{i}$ to $X_{\pi(i)}(1\leqq i\leqq k)$

Remark 0.3. $\rho_{k+1}$ can be extended to an automorphism $\rho_{k+1}^{*}$ of $R_{k}$ by $\rho_{k+1}^{*}(X_{j})=$

Now,

assume

further that there exist elements $u_{i}(1\leqq i\leqq e)$ in $B$ such that

(iv) $bu_{i}=u_{i}\rho_{i}^{m_{l}}(b)(b\in B)$

and

(v) $\rho_{j}(u_{i})u_{ji}\rho_{i}(u_{ji})\cdots\rho_{i}^{m-1}:(u_{ji})=u_{i}(1\leqq i\leqq e)$

.

Thenwe have,

$a(X_{i}^{m:}-u_{i})=(X_{i}^{m:}-\%)\rho_{i}^{m}$‘$(a)(a\in B)$ and

$X_{j}(X_{i}^{m_{l}}-u_{i})=(X_{i}^{m\prime}-\%)X_{j}u_{ji}\rho_{i}(u_{ji})\cdots\rho_{i}^{m_{*}-1}(u_{ji})(1\leqq i,j\leqq e)$

.

This

means

$(X_{i}^{m:}-\%)R_{k}$ is a two-sided ideal of $R_{k}$ for _{$i\leqq k\leqq e$}

.

The mapping

$\overline{\rho}_{i}$

:

$R_{e}arrow R_{e}$ defined by

$\overline{\rho}_{i}(\sum X_{1^{1}}^{\nu}X_{2^{2}}^{\nu}\cdots X_{e}^{\nu_{*}}b_{\nu_{1}\nu_{2}\cdots\nu_{e}})=\sum(X_{1}u_{1i})^{\nu_{1}}(X_{2}u_{2i})^{\nu_{2}}\cdots(X_{e}u_{\dot{a}})^{\nu_{\epsilon}}\rho_{i}(b_{\nu_{1}\nu_{2}\cdots\nu_{e}})$

is

an

automorphism of$R_{e}$ which is

an

extension of

$\rho_{i}$

.

We put here

$B_{i}=B[X_{1}, \cdots X_{i-1}, X_{i+1}, \cdots X_{e};\rho_{1}, \cdots\rho_{i-1},\rho_{i+1}, \cdots\rho_{e};\{u_{ij}\}]$

.

$Naturag_{y}$

we

have

$R_{e}=B_{i}[X_{i};\overline{\rho}_{i}]$, and

$\beta(X_{i}^{m_{i}}-u_{i})=(X_{i}^{m}:-u_{i})\overline{\rho}_{i}^{m_{i}}(\beta)(\beta\in B_{i})$ and $\overline{\rho}_{i}(u_{t})=u_{i}$,

where $\overline{\rho}_{i}$

means

$\overline{\rho}_{i}|B_{i}$

.

Let $M=(X_{1}^{m_{1}}-u_{1}, X_{2}^{m_{2}}, \cdots X_{e}^{m_{e}}-u_{e})$ be the two sided ideal of$R_{e}$ generatedby

$\{X_{1}^{m_{1}}-u_{1}, X_{2}^{m_{2}}, \cdots , X_{e}^{m_{C}}-u_{e}\}$

.

Thenthe factor ring $R_{e}/M$ is a free ringextension

over

$B$ with a basis

$\{x_{1}^{\nu_{1}}x_{2^{2}}^{\nu}\cdots x_{e}^{\nu_{e}}|0\leqq\nu_{i}<m, 1\leqq i\leqq e\}$, where $x_{i}=X_{i}+M\in R_{e}/M$

.

Under the above notations,

we

shall state the $f_{0}g_{oW}ing$ theorem which is a

gener-alization of Proposition 1.

Theorem 3. The following are equivalent.

(1) $R_{e}/M$ is a separable extension

of

$B$

.

(2) (a) $u_{i}\in U(B^{\rho}$‘$)$ $(1 \leqq i\leqq e)$

.

(b) There ezists an element $z\in Z$ such that

$\sum_{i=10}^{e}\sum_{\leqq\nu_{i}<m_{i}}\rho_{1}^{\nu_{1}}\rho_{2}^{\nu_{2}}\cdots\rho_{e}^{\nu_{e}}(z)=1$

(3) $X_{i}^{m_{*}}-u_{i}$ is a separable polynomial in $B_{i}[X_{i};\overline{\rho}_{i}]$

for

each $i(1\leqq i\leqq e)$

.

(4) (a) %\in U(B ) $(1 \leqq i\leqq e)$

.

(b) There

evtst

elements $q\in Z^{p_{1},\rho_{2},\cdots,\rho_{i-1},\rho+1}$ such that

To state the result concerning H-sesparable extensions,

we

need

some more

nota-tions.

$S_{0}=B$ and $S_{1}=B[X_{1};\rho_{1}]/(X_{1}^{m_{1}}-u_{1})B[X_{1}; \rho_{1}]$

.

For $1\leqq k\leqq e$, We put here $S_{k}=S_{k-1}[X_{k};\overline{\rho}_{k}]/(X_{k}^{m_{k}}-u_{k})S_{k-1}[X_{k};\overline{\rho}_{k}]$

.

$Naturag_{y}$ we have

$R_{e}/M=S_{e}\supset S_{e-1}\supset\cdots\supset S_{1}\supset S_{0}=B$

.

Under the above notations,

we

have the following:

Theorem 4. Thefollounng are equivalent.

(1) $R./M$ is an H-separable extension

_of

$B$, and the centralizers

of

$B$ in _{$R_{e}/M$},

$V_{R_{*}/M}(B)=Z$

.

(2) $X_{k}^{m_{k}}-u_{k}$ is an H-separable polynomial in $S_{k-1}[X_{k};\overline{\rho}_{k}]$

for

each$k(1\leqq k\leqq e)$

.

(3) (a) $\mathfrak{R}\in U(B^{\rho:})$ $(1 \leqq i\leqq e)$

(b) The oder

_of

$(\rho_{i}|Z)=m(1\leqq i\leqq e)$, the set$\{\rho_{i}|Z|1\leqq i\leqq e\}$ genemtes

an abelian group $<\rho_{1}|Z>\cross<\rho_{2}|Z>\cross\cdots\cross<\rho_{e}|Z>=G$, and $Z/Z^{G}$

is a G-Galois extension.

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