On the structure of skew groupoid rings which are Azumaya
In this paper we present an intrinsic description of the structure of an Azumaya skew groupoid ring, having its center contained in the respective ground ring, in terms of suitable central Galois algebras and commutative Galois extensions.
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irk-123456789-1523092019-06-10T01:26:21Z On the structure of skew groupoid rings which are Azumaya Flores, D. Paques, A. In this paper we present an intrinsic description of the structure of an Azumaya skew groupoid ring, having its center contained in the respective ground ring, in terms of suitable central Galois algebras and commutative Galois extensions. 2013 Article On the structure of skew groupoid rings which are Azumaya / D. Flores, A. Paques // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 71–80. — Бібліогр.: 15 назв. — англ. 1726-3255 2010 MSC:16H05, 18B40, 20L99. http://dspace.nbuv.gov.ua/handle/123456789/152309 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper we present an intrinsic description of the structure of an Azumaya skew groupoid ring, having its center contained in the respective ground ring, in terms of suitable central Galois algebras and commutative Galois extensions. |
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On the structure of skew groupoid rings which are Azumaya |
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On the structure of skew groupoid rings which are Azumaya |
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On the structure of skew groupoid rings which are Azumaya |
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On the structure of skew groupoid rings which are Azumaya / D. Flores, A. Paques // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 71–80. — Бібліогр.: 15 назв. — англ. |
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Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 16 (2013). Number 1. pp. 71 – 80
© Journal “Algebra and Discrete Mathematics”
On the structure of skew groupoid rings
which are Azumaya
Daiana Flôres, Antonio Paques
Communicated by V. V. Kirichenko
Abstract. In this paper we present an intrinsic description
of the structure of an Azumaya skew groupoid ring, having its center
contained in the respective ground ring, in terms of suitable central
Galois algebras and commutative Galois extensions.
1. Introduction
Groupoids are usually presented as small categories whose morphisms
are invertible.
The notion of a groupoid action that we use in this paper arose from
the notion of a partial groupoid action [3], which is a natural extension
of the notion of a partial group action [5]. First, partial ordered groupoid
actions on sets were introduced in the literature, as ordered premorphisms,
by N. Gilbert [8]. After, partial ordered groupoid actions on rings were
considered by D. Bagio and the authors [2] as a generalization of partial
group actions, as introduced by M. Dokuchaev and R. Exel in [5]. And in
[3] this notion was extended to the general context of groupoids.
Accordingly [3], an action of a groupoid G on a K-algebra R is a pair
β = ({Eg}g∈G, {βg}g∈G), where for each g ∈ G, Eg = Er(g) is an ideal of
R and βg : Eg−1 → Eg is an isomorphism of K-algebras satisfying some
suitable conditions of compatibility (see the subsection 2.3).
2010 MSC: 16H05, 18B40, 20L99.
Key words and phrases: groupoid action, skew groupoid ring, Azumaya ring,
Galois algebra.
72 On the structure of skew groupoid rings
The notion of groupoid action given by Caenepeel and De Groot in [4]
is equivalent to the above one in the case that the set G0 of all identities
of G is finite, each ideal Ee (e ∈ G0) is unital and R =
⊕
e∈G0
Ee, [7,
Proposition 2.2]. This is the groupoid action we will deal with throughout
all this paper.
Given such an action β of G on R, we can consider the skew groupoid
ring A = R ⋆β G, similarly to the construction given in [2], which is an
associative and unital K-algebra.
Our aim is to present an intrinsic structural description of A in the
case that G is finite, A is Azumaya and its center is contained in R (see
Theorem 3.3).
This paper is organized as follows. In the next section we give pre-
liminaries about groupoids, groupoid actions, skew groupoid rings, and
separability, Hirata-separability, Galois, and Azumaya properties. In that
section we will be concerned only with the results strictly necessary to
construct the appropriate conditions to prove our main theorem, whose
proof we set in the section 3.
Throughout, unless otherwise specified, rings and algebras are asso-
ciative and unital.
2. Definitions and basic results
2.1. Groupoids
The axiomatic version of groupoid that we adopt in this paper was
taken from [13]. A groupoid is a non-empty set G equipped with a partially
defined binary operation, that we will denote by concatenation, for which
the usual axioms of a group hold whenever they make sense, that is:
(i) For every g, h, l ∈ G, g(hl) exists if and only if (gh)l exists and in
this case they are equal.
(ii) For every g, h, l ∈ G, g(hl) exists if and only if gh and hl exist.
(iii) For each g ∈ G there exist (unique) elements d(g), r(g) ∈ G such
that gd(g) and r(g)g exist and gd(g) = g = r(g)g.
(iv) For each g ∈ G there exists an element g−1 ∈ G such that d(g) =
g−1g and r(g) = gg−1.
We will denote by G2 the subset of the pairs (g, h) ∈ G × G such that
the element gh exists.
D. Flôres, A. Paques 73
An element e ∈ G is called an identity of G if e = d(g) = r(g−1), for
some g ∈ G. In this case e is called the domain identity of g and the range
identity of g−1. We will denote by G0 the set of all identities of G and we
will denote by Ge the set of all g ∈ G such that d(g) = r(g) = e. Clearly,
Ge is a group, called the isotropy (or principal) group associated to e.
The assertions listed in the following lemma are straightforward from
the above definition. Such assertions will be freely used along this paper.
Lemma 2.1. Let G be a groupoid. Then,
(i) for every g ∈ G, the element g−1 is unique satisfying g−1g = d(g)
and gg−1 = r(g),
(ii) for every g ∈ G, d(g−1) = r(g) and r(g−1) = d(g),
(iii) for every g ∈ G, (g−1)−1 = g,
(iv) for every g, h ∈ G, (g, h) ∈ G2 if and only if d(g) = r(h),
(v) for every g, h ∈ G, (h−1, g−1) ∈ G2 if and only if (g, h) ∈ G2 and,
in this case, (gh)−1 = h−1g−1,
(vi) for every (g, h) ∈ G2, d(gh) = d(h) and r(gh) = r(g),
(vii) for every e ∈ G0, d(e) = r(e) = e and e−1 = e,
(viii) for every (g, h) ∈ G2, gh ∈ G0 if and only if g = h−1,
(ix) for every g, h ∈ G, there exists l ∈ G such that g = hl if and only
if r(g) = r(h),
(x) for every g, h ∈ G, there exists l ∈ G such that g = lh if and only
if d(g) = d(h).
2.2. Groupoid actions and skew groupoid rings
Let G be a groupoid and R an algebra over a commutative ring K.
Following [3], an action of G on R is a pair
β = ({Eg}g∈G, {βg}g∈G),
where for each g ∈ G, Eg = Er(g) is an ideal of R, βg : Eg−1 → Eg is
an isomorphism of K-algebras (not necessarily unital), and the following
conditions hold:
(i) βe is the identity map IEe of Ee, for all e ∈ G0,
(ii) βgβh(r) = βgh(r), for all (g, h) ∈ G2 and r ∈ Eh−1 = E(gh)−1 .
74 On the structure of skew groupoid rings
In particular, β induces an action of the group Ge on Ee, for every e ∈ G0.
Accordingly [2, Section 3], the skew groupoid ring R⋆β G corresponding
to β is defined as the direct sum
R ⋆β G =
⊕
g∈G
Egδg
in which the δg’s are symbols, with the usual addition, and multiplication
determined by the rule
(xδg)(yδh) =
{
xβg(y)δgh if (g, h) ∈ G2
0 otherwise,
for all g, h ∈ G, x ∈ Eg and y ∈ Eh.
This multiplication is well defined. Indeed, if (g, h) ∈ G2 then d(g) =
r(h) (see Lemma 2.1(iv)). So, Eg−1 = Er(g−1) = Ed(g) = Er(h) = Eh,
βg(y) makes sense, and xβg(y) ∈ Eg = Er(g)
(⋆)
= Er(gh) = Egh, where the
equality (⋆) is ensured by Lemma 2.1(vi).
By a routine calculation one easily sees that A = R ⋆β G is associative,
and by [2, Proposition 3.3] it is unital if G0 is finite and Ee is unital, for
all e ∈ G0. In this case, the identity element of A is 1A =
∑
e∈G0
1eδe,
where 1e denotes the identity element of Ee, for all e ∈ G0.
2.3. Invariants, Galois extensions and Galois algebras
Let R, G and β as in the preceding subsection.
For our purposes we will assume hereafter that G is finite, R =
⊕
e∈G0
Ee and every ideal Eg is unital, with its identity element denoted
by 1g (in particular, each 1g is a central idempotent of R).
We will denote by Rβ the subalgebra of the invariant elements of R
under the action β, that is,
Rβ = {r ∈ R | βg(r1g−1) = r1g, for all g ∈ G}.
For any non-empty subset X of R, any subring Y of R containing X
and any (Y, Y )-bimodule V we will denote by CV (X) the centralizer of
X in V , that is, the set of all v ∈ V such that xv = vx for all x ∈ X. If,
in particular, X = Y = V = R, then CV (X) is the center of R and we
will denote it simply C(R).
A non-empty subset X of R is called β-invariant if βg(Eg−1 ∩ X) =
Eg ∩ X, for every g ∈ G.
D. Flôres, A. Paques 75
In particular, C(R) and CR(Rβ) are β-invariant. Indeed, if r ∈ Eg−1 ∩
C(R), then
βg(r)x = βg(r)1gx = βg(rβg−1(x1g)) =
= βg(βg−1(x1g)r) = x1gβg(r) = xβg(r),
for all x ∈ R, and so βg(r) ∈ Eg ∩ C(R), for all g ∈ G. Hence, βg(Eg−1 ∩
C(R)) = Eg ∩ C(R). By similar arguments one also gets βg(Eg−1 ∩
CR(Rβ)) = Eg ∩ CR(Rβ).
Moreover, since 1g ∈ C(R) ⊆ CR(Rβ), β induces by restriction an
action on X ′ = CR(X), with X = R or Rβ, given by the pair β|X′ =
({E′
g = X ′1g}g∈G, {βg|E′
g−1
}g∈G). Notice that E′
g = E′
r(g), for all g ∈ G,
and X ′ =
⊕
e∈G0
E′
e.
We say that R is a β-Galois extension (resp., G-Galois extension, if G
is a group) of Rβ (resp., RG) if there exist elements xi, yi ∈ R, 1 ≤ i ≤ m,
such that
∑
1≤i≤m
xiβg(yi1g−1) = δe,g1e,
for all e ∈ G0 and g ∈ G. In particular, in this case, Ee is a Ge-Galois
extension of EGe
e for every e ∈ G0. The set {xi, yi | 1 ≤ i ≤ n} is called a
Galois coordinate system of R over Rβ (resp., RG).
We also say that R is a β-Galois algebra (resp., β-central Galois algebra)
if R is a β-Galois extension of Rβ and Rβ ⊆ C(R) (resp., Rβ = C(R)).
In the particular case that G is a group, we replace “β-" by “G-".
Remark 2.2. Observe that Rβ ⊆
⊕
EGe
e . Indeed, any element x ∈ R
is of the form x =
∑
e∈G0
xe, with xe ∈ Ee, and x ∈ Rβ if and only
if βg(x1g−1) = x1g if and only if βg(xd(g)) = xr(g), for all g ∈ G. In
particular, βg(xe) = xe, for all e ∈ G0 and g ∈ Ge. Hence, each xe belongs
to EGe
e and x ∈
⊕
e∈G0
EGe
e .
In general the inclusion Rβ ⊆
⊕
EGe
e is strict, as it is shown in
the following example: take G = {g, g−1, r(g), d(g)}, R = Se1 ⊕ Se2 ⊕
Se3 ⊕ Se4, where S is a ring and e1, e2, e3 and e4 are pairwise orthogonal
idempotents with sum 1R, Eg = Er(g) = Se1 ⊕ Se2, Eg−1 = Ed(g) =
Se3 ⊕ Se4, βd(g) = IEd(g)
, βr(g) = IEr(g)
, βg(xe3 + ye4) = xe1 + ye2
and βg−1(xe1 + ye2) = xe3 + ye4, para todo x, y ∈ S. It is immediate
to check that β = ({El}l∈G, {βl}l∈G) is an action of G on R, Rβ =
S(e1 + e3) ⊕ S(e2 + e4), E
Gd(g)
d(g) = Se3 ⊕ Se4 and E
Gr(g)
r(g) = Se1 ⊕ Se2. In
this example we also notice that Rβ1d(g) = E
Gd(g)
d(g) and Rβ1r(g) = E
Gr(g)
r(g) .
76 On the structure of skew groupoid rings
But, neither this fact occurs in general, as we see in the next example:
take R as above and G = {g1, g2, g}, with G0 = {g1, g2}, g−1 = g and
gg = g2. Setting Eg1 = Se1 ⊕ Se4, Eg2 = Se2 ⊕ Se3 = Ed(g) = Er(g) and
βg(ae2 +be3) = be2 +ae3, one easily sees that Rβ = Se1 ⊕S(e2 +e3)⊕Se4
and Rβ1g2 = S(e2 + e3) ⊂ Eg2 = E
Gg2
g2 .
2.4. Azumaya, Hirata-separable and separable algebras
Let S be a subalgebra of R. We say that R is separable over S [11]
if R is a direct summand of R ⊗S R as an R-bimodule. In this case, if
S ⊆ C(R) (resp., S = C(R)) we also say that R is a separable S-algebra
(resp., Azumaya over S or Azumaya algebra or simply Azumaya). We
say that R is Hirata-separable over S [9] if R ⊗S R is isomorphic, as an
R-bimodule, to a direct summand of a finite direct sum of copies of R. It
is well known that every Azumaya algebra is Hirata-separable over its
center [14] as well as every Hirata-separable extension is separable [10].
3. The main results
Also in this section, R, G and β are as in the later one.
We start with the following theorem, which is a generalization of
[1, Theorems 1 and 2] to the setting of groupoid actions, and shows a
nice and closed relation among the notions of Azumaya algebra, Galois
extension and Hirata-separability. The arguments used in the proof of
the mentioned theorems in [1] are practically the same we will use here.
Theorem 3.1. The following statements are equivalent:
(i) A is Azumaya and C(A) ⊆ R.
(ii) A is Hirata-separable over R and R is separable over C(R)β.
(iii) R is a β-Galois extension of Rβ and Rβ is Azumaya over C(A).
(iv) CR(Rβ) is a β-Galois extension of C(A) and Rβ is Azumaya over
C(A).
Moreover, in this case, C(A) = C(R)β = C(Rβ).
For the proof of this theorem we need the following lema.
Lemma 3.2. C(A) ⊆ R if and only if C(A) = C(R)β.
D. Flôres, A. Paques 77
Proof. If C(A) ⊆ R, then C(A) ⊆ C(R) and (
∑
e∈G0
x1eδe)1gδg =
1gδg(
∑
e∈G0
x1eδe), for every x ∈ C(A) and g ∈ G.
Since (
∑
e∈G0
x1eδe)1gδg = (x1r(g)δr(g))(1gδg) = x1gδg and
1gδg(
∑
e∈G0
x1eδe) = (1gδg)(x1d(g)δd(g)) = βg(x1g−1)δg, it follows
that βg(x1g−1) = x1g, for every g ∈ G. Therefore x ∈ C(R)β. The
converse is straightforward.
Proof of Theorem 3.1.
(i)⇒(ii) By Lemma 3.2 we have C(A) = C(R)β. It follows from
[12, Theorem 1] that A is Hirata-separable over R, since A is a finitely
generated projective R-module. In addition, R is a direct summand of
A as an R-bimodule. Hence, it follows from [14, Proposition 1.3] that
CA(R) is separable over C(A) and CA(CA(R)) = R. Consequently, R is
separable over C(A) by [6, Theorem III.4.3].
(ii)⇒(iii) We will proceed by steps.
step 1: A is Azumaya and C(A) ⊆ R.
Since A is Hirata-separable over R and R is a direct summand of
A as an R-bimodule, it follows that CA(R) is separable over C(A) and
CA(CA(R)) = R, by [14, Proposition 1.3]. Furthermore, C(A) = C(R)β by
Lemma 3.2. On the other hand, by the assumptions and [10, Theorem 2.2]
we have that A is separable over R, so over C(R)β too.
step 2: E = EndC(R)β (R) is Azumaya over C(R)β .
Indeed, A is Azumaya over C(R)β (by step 1) and so a projective
finitely generated C(R)β-module, by [6, Theorem III.3.4]. Hence, R also
is a projective finitely generated C(R)β-module, as a direct summand of
A. Now, the assertion follows by [6, Theorem III.4.1].
step 3: CE(A) = EndA(R) ≃ (Rβ)op as C(R)β-algebras.
First of all, we observe that R is a left A-module via the action given
by rgδg · r = rgβg(r1g−1), for all g ∈ G, rg ∈ Eg and r ∈ R. Consequently,
E is an A-bimodule in a natural way, that is, af(r) = a · f(r) and
fa(r) = f(a · r), for all a ∈ A, f ∈ E and r ∈ R. Now, the equality of
the assertion is immediate. It is also immediate that R is a ((Rβ)op, A)-
bimodule, with (Rβ)op acting on R by the left via the right multiplication,
that is, x · r = rx, for all r ∈ R and x ∈ (Rβ)op.
The claimed isomorphism of the assertion is given by the map θ :
(Rβ)op → EndA(R) defined by θ(x)(r) = x · r, for all x ∈ (Rβ)op and
r ∈ R, whose inverse is given by f 7→ f(1R), for all f ∈ EndA(R).
Now, we are in condition to prove the statement (iii).
78 On the structure of skew groupoid rings
It follows from the assumptions, step 3 and [12, Lemma 1] that R is a
generator in the category of the left A-modules. Hence, R is a β-Galois
extension of Rβ, as well as A ≃ End(R)Rβ as C(R)β-algebras, by [3,
Theorem 3]. Then, A can be seen as a C(R)β-subalgebra of E and, by
steps 1 and 2 and [6, Theorem III.4.3], we have that CE(A) is Azumaya
over C(R)β = C(A). So, (Rβ)op (and, therefore, also Rβ) is Azumaya over
C(R)β by step 3, which also ensure that C(Rβ) = C(R)β .
(iii)⇒(i) By [3, Theorem 3] we have that R is a projective finitely
generated right Rβ-module, A ≃ End(R)Rβ as C(R)β-algebras. By [6,
Theorem III.3.4] we have that Rβ is a projective finitely generated
C(A)(= C(Rβ))-module. By Lemma 3.2, C(A) = C(R)β. Therefore,
R is a projective finitely generated C(R)β-module. Moreover, R is a gen-
erator in the category of the C(R)β-modules. Hence, E = EndC(R)β (R)
is Azumaya over C(R)β by [6, Theorem III.4.1], and CE(Rβ) is Azumaya
over C(R)β by [6, Theorem III.4.3].
Finally, End(R)Rβ = CE(Rβ). Indeed, it is enough to notice that
End(R)Rβ is a subalgebra of E and a Rβ-bimodule via the actions
xfy(r) = f(rx)y, for all f ∈ End(R)Rβ , x, y ∈ Rβ and r ∈ R.
(iii)⇒(iv) By assumption Rβ is Azumaya over C(A) and it follows
from the equivalence (iii)⇔(i) that A is Azumaya. By [6, Theorem III.4.3],
CA(Rβ) is Azumaya over C(A).
Therefore, A′ = CR(Rβ) ⋆β G is Azumaya and C(A′) = C(A) =
C(Rβ) ⊆ CR(Rβ). Then, it follows again from the equivalence (i)⇔(iii)
that CR(Rβ) is a β-Galois extension of CR(Rβ)β = C(Rβ) = C(A).
(iv)⇒(iii) It is enough to observe that, by assumption, there exist ele-
ments xi, yi ∈ CR(Rβ) ⊆ R, 1 ≤ i ≤ m, such that
∑
1≤i≤m xiβg(yi1g−1) =
δe,g1e, for all g ∈ G and e ∈ G0.
Theorem 3.3. Assume that A is Azumaya, C(A) ⊆ R, and R′ = CR(Rβ)
satisfies R′β1e = (E′
e)Ge, where E′
e = R′1e, for all e ∈ G0. Then,
(i) A ≃ Rβ ⊗C(A) (R′ ⋆β G) as C(A)-algebras.
(ii) R′ ⋆β G ≃ End(R′)R′β as C(A)-algebras.
(iii) R′ =
⊕
e∈G0
E′
e.
(iv) for each e ∈ G0 there exist pairwise orthogonal idempotents ve,i ∈
C(E′
e), 1 ≤ i ≤ me, and subgroups Hi of Ge such that
– each E′
eve,i is an Hi-central Galois algebra,
D. Flôres, A. Paques 79
– E′
e =
⊕
1≤i≤me
E′
eve,i or E′
e = (
⊕
1≤i≤me
E′
eve,i)
⊕
C(E′
e)ve, with
ve = 1e −
∑
1≤i≤me
ve,i,
– Ge|C(E′
e)ve
≃ Ge and
– C(E′
e)ve = E′
eve is a commutative Galois extension of (E′
eve)Ge .
Proof. (i) It follows from Theorem 3.1 and [6, Theorem III.4.3] that
A ≃ Rβ ⊗C(A) CR(Rβ) as C(A)-algebras. It is routine to check that
CA(Rβ) = R′ ⋆β G.
(ii) It follows from Theorem 3.1(vi) that R′ is a β-Galois extension
of C(A) = C(Rβ) = R′β. So, R′ ⋆β G ≃ End(R′)R′β as C(A)-algebras.
(iii) It is immediate, since R =
⊕
e∈G0
Ee.
(iv) It follows from Theorem 3.1 and [6, Theorem III.4.3] that R′⋆βG =
CA(Rβ) is Azumaya over C(A), which implies that C(R′ ⋆β G) = C(A) =
C(Rβ) = R′β and, consequently, R′β ⊆ C(R′). Thus, (E′
e)
Ge = R′β1e ⊆
C(R′)1e = C(R′1e) = C(E′
e), that is, E′
e is a Ge-Galois algebra for all
e ∈ G0. The result follows now from [15, Theorem 3.8].
References
[1] R. Alfaro and G. Szeto, Skew group rings which are Azumaya, Comm. Algebra 23
(1995), 2255-2261.
[2] D. Bagio, D. Flôres and A. Paques, Partial actions of ordered groupoids on rings,
J. Algebra Appl. 9 (2010), 501-517.
[3] D. Bagio and A. Paques, Partial groupoid actions: globalization, Morita theory
and Galois theory, Comm. Algebra 40 (2012), 3658-3678.
[4] S. Caenepeel and E. De Groot, Galois theory for weak Hopf algebras, Rev. Roumaine
Math. Pures Appl. 52 (2007), 151-176.
[5] M. Dokuchaev and R. Exel, Associativity of crossed products by partial actions,
enveloping actions and partial representations, Trans. AMS 357 (2005), 1931-1952.
[6] F. DeMeyer and E. Ingraham, Separable Algebras over Commutative Rings, LNM
181, Springer Verlag, Berlin, 1971.
[7] D. Flôres and A. Paques, Duality for groupoid (co)actions, to appear in Comm.
Algebra (2013).
[8] N.D. Gilbert, Actions and expansions of ordered groupoids, J. Pure Appl. Algebra
198 (2005), 175-195.
[9] M. Harada, Suplementary results on Galois extensions, Osaka J. Math. 2 (1965),
287 - 295.
[10] K. Hirata, Some types of separable extensions of rings, Nagoya Math. J. 33 (1968),
107-115.
[11] K. Hirata and K. Sugano, On semisimple extensions and separable extensions over
noncommutative rings, J. Math. Soc. Japan 18 (1966), 360 - 373.
80 On the structure of skew groupoid rings
[12] S. Ikehata, Note on Azumaya algebras and H-separable extensions, Math. J.
Okayama Univ. 23 (1981), 17-18.
[13] M.V. Lawson, Inverse Semigroups. The Theory of Partial Symmetries, World
Scientific Pub. Co, London, 1998.
[14] K. Sugano, Note on semisimple extensions and separable extensions, Osaka J.
Math. 4 (1967), 265-270.
[15] G. Szeto and L. Xue. The structure of Galois Algebras, J. Algebra 237 (2001),
238-246.
Contact information
Daiana Flôres Departamento de Matemática, Universidade
Federal de Santa Maria, 97105-900, Santa Maria,
RS, Brazil
E-Mail: flores@ufsm.br
Antonio Paques Instituto de Matemática, Universidade Federal
do Rio Grande do Sul, 91509-900, Porto Alegre,
RS, Brazil
E-Mail: paques@mat.ufrgs.br
Received by the editors: 11.04.2013
and in final form 11.04.2013.
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