A Morita context related to finite groups acting partially on a ring
In this paper we consider partial actions of groups on rings, partial skew group rings and partial fixed rings. We study a Morita context associated to these rings, α-partial Galois extensions and related aspects. Finally, we establish conditions to obtain a Morita equivalence between Rα and R⋆αG....
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Інститут прикладної математики і механіки НАН України
2009
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| Цитувати: | A Morita context related to finite groups acting partially on a ring/ J.A. Guzman, J. Lazzarin // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 49–60. — Бібліогр.: 10 назв. — англ. |
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irk-123456789-1546302019-06-16T01:29:30Z A Morita context related to finite groups acting partially on a ring Guzman, J.A. Lazzarin, J. In this paper we consider partial actions of groups on rings, partial skew group rings and partial fixed rings. We study a Morita context associated to these rings, α-partial Galois extensions and related aspects. Finally, we establish conditions to obtain a Morita equivalence between Rα and R⋆αG. 2009 Article A Morita context related to finite groups acting partially on a ring/ J.A. Guzman, J. Lazzarin // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 49–60. — Бібліогр.: 10 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16S35, 16R30, 13C60, 16N60. http://dspace.nbuv.gov.ua/handle/123456789/154630 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper we consider partial actions of groups on rings, partial skew group rings and partial fixed rings. We study a Morita context associated to these rings, α-partial Galois extensions and related aspects. Finally, we establish conditions to obtain a Morita equivalence between Rα and R⋆αG. |
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Article |
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Guzman, J.A. Lazzarin, J. |
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Guzman, J.A. Lazzarin, J. A Morita context related to finite groups acting partially on a ring Algebra and Discrete Mathematics |
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Guzman, J.A. Lazzarin, J. |
| author_sort |
Guzman, J.A. |
| title |
A Morita context related to finite groups acting partially on a ring |
| title_short |
A Morita context related to finite groups acting partially on a ring |
| title_full |
A Morita context related to finite groups acting partially on a ring |
| title_fullStr |
A Morita context related to finite groups acting partially on a ring |
| title_full_unstemmed |
A Morita context related to finite groups acting partially on a ring |
| title_sort |
morita context related to finite groups acting partially on a ring |
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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/123456789/154630 |
| citation_txt |
A Morita context related to finite groups acting partially on a ring/ J.A. Guzman, J. Lazzarin // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 49–60. — Бібліогр.: 10 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT guzmanja amoritacontextrelatedtofinitegroupsactingpartiallyonaring AT lazzarinj amoritacontextrelatedtofinitegroupsactingpartiallyonaring AT guzmanja moritacontextrelatedtofinitegroupsactingpartiallyonaring AT lazzarinj moritacontextrelatedtofinitegroupsactingpartiallyonaring |
| first_indexed |
2025-07-14T06:40:24Z |
| last_indexed |
2025-07-14T06:40:24Z |
| _version_ |
1837603461469831168 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 3. (2009). pp. 49 – 60
c⃝ Journal “Algebra and Discrete Mathematics”
A Morita context related to finite groups acting
partially on a ring
Jesús Ávila Guzmán and João Lazzarin
Communicated by guest editors
Abstract. In this paper we consider partial actions of groups
on rings, partial skew group rings and partial fixed rings. We study
a Morita context associated to these rings, �-partial Galois exten-
sions and related aspects. Finally, we establish conditions to obtain
a Morita equivalence between R� and R ★� G.
Dedicated to Professor Miguel Ferrero
on occasion of his 70-th anniversary
Introduction
Partial actions of groups have been introduced in the theory of oper-
ator algebras giving powerful tools of their study (see [4], [6], [5], [2] and
the literature quoted therein). Also in [4] the authors introduced partial
actions on algebras in a pure algebraic context. Let G be a group and
R a unital k-algebra, where k is a commutative ring. A partial action �
of G on R is a collection of ideals Dg, g ∈ G, of R and isomorphisms of
(non-necessarily unital) k -algebras �g : Dg−1 → Dg such that:
(i) D1 = R and �1 is the identity mapping of R;
(ii) D(gℎ)−1 ⊇ �−1
ℎ (Dℎ ∩Dg−1), for any g, ℎ ∈ G;
Both authors were partially supported by Conselho Nacional de Desenvolvimento
Cient́ıfico e Tecnológico (CNPq, Brazil).
2000 Mathematics Subject Classification: 16S35, 16R30, 13C60, 16N60 .
Key words and phrases: partial action, skew group ring, fixed ring, Morita
context, Morita equivalence, semiprime ring.
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.50 Morita context and partial actions of groups on rings
(iii) �g ∘ �ℎ(x) = �gℎ(x), for any x ∈ �−1
ℎ (Dℎ ∩Dg−1) and g, ℎ ∈ G.
Using (iii) we can easily see that �g−1 = �−1
g , for every g ∈ G. Also
the property (ii) can be written as �g(Dg−1 ∩ Dℎ) = Dg ∩ Dgℎ, for all
g, ℎ ∈ G.
Let � be a partial action of G on R. The partial skew group ring
S = R ★� G (see [4]) is defined as the set of all finite formal sums
∑
g∈G ag�g, ag ∈ Dg for every g ∈ G, where the addition is defined
in the usual way and the multiplication is determined by (ag�g)(bℎ�ℎ) =
�g(�g−1(ag)bℎ)�gℎ.
Given a partial action � of a group G on R an enveloping action for
� is an algebra T together with a global action � = {�g ∣ g ∈ G} of
G on T , where each �g is an automorphism of T , such that the partial
action is given by restriction of the global action (see [4] and [6] for more
precise definition and properties). From Theorem 4.5 of [4] we know that
a partial action � has an enveloping action if and only if all the ideals
Dg are unital algebras, i.e., Dg is generated by a central idempotent of
R, for any g ∈ G. In this case the partial skew group ring R ★� G is an
associative algebra, which is not true in general (see [4], Example 3.5).
Throughout this paper R is an associative k-algebra (which will be
called frequently simply a ring) with an identity element 1R, G is a finite
group and � = {�g : Dg−1 → Dg} is a partial action of G on R. We
assume, unless otherwise stated, that the partial action has an enveloping
action denoted by (T, �). Then any of the ideals Dg is generated by a
central idempotent of R which we denote by 1g. Since that (T, �) is the
enveloping action of (R,�) we have that 1g = 1R�g(1R) where �g(1R) are
central elements in T for every g ∈ G. These facts will be used freely in
this paper.
In general, T does not need to have an identity element; but it has
an identity when G is a finite group. In this case, the fixed ring of T
will be denoted by TG and the trace map by trG =
∑
g∈G g. The ring
of the invariant elements of R under � (the partial fixed ring) is R� =
{
x ∈ R : �g(x1g−1) = x1g, for any g ∈ G
}
and the partial trace map is
defined by tr� (r) =
∑
g∈G �g
(
r1g−1
)
, for any r ∈ R (see [5] and [7] for
details).
In [5], Dokuchaev, Ferrero and Paques introduced the notion of par-
tial Galois extension and developed a Galois theory for partial actions.
The existence of partial Galois coordinates introduced in [5] is necessary
(but not sufficient) to establish a Morita equivalence between the partial
fixed ring and the partial skew group ring. In this paper, among other
results, we show some applications of these concepts. In the first Sec-
tion, following the global case (see [3]), we establish a Morita context
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.J. Guzmán, J. Lazzarin 51
(R�, S = R ★� G, V,W,Γ,Γ′). In Section 2 we study the non-degeneracy
of Γ and Γ′ and some consequences. In Section 3 we show, under the
assumption that the partial trace map is onto, that R is an �-partial
Galois extension of R� if and only if the Morita Context given is strict,
and in this case, R� and S are Morita equivalent rings. Finally, Section 4
is devoted to establish some class of rings that are each one an �-partial
Galois extension of its corresponding partial fixed subring.
1. A Morita context for R
� and R ★� G
Following the global case ([3] and [1]), we will construct the partial version
of a Morita context, that is, the six-tuple (R�, S = R ★� G, V,W,Γ,Γ′)
where V = R�RS , W = SRR� and Γ : V ⊗SW → R� and Γ′ : W⊗R�V →
S are defined by
Γ(x⊗ y) = tr�(xy) =
∑
g∈G
�g(xy1g−1), (1)
Γ′(x⊗ y) =
∑
g∈G
x�g(y1g−1)�g, (2)
for all x, y ∈ R. For this we need some preparation.
First of all, it is clear that R has a structure of a (R�, R�)-bimodule
via the multiplication of R and it is easy to check that R is a (S,R�)-
bimodule (resp. (R�, S)-bimodule) with the left (resp. right) action of S
on R given by a�g ⋅ r = a�g(r1g−1) (resp. r ⋅ a�g = �g−1(ra)), for every
g ∈ G, a ∈ Dg and r ∈ R.
In order to prove that Γ and Γ′ are well defined we need the following
auxiliary result which is trivial in the global case.
1.1. Lemma. tr� (x) = tr�(�g(x)), for any g ∈ G and x ∈ Dg−1.
Proof. First, note that �g(1g−11ℎ) = 1g1gℎ and �ℎ(�g(x1g−1)1ℎ−1) =
�ℎg
(
x1g−1ℎ−1
)
1ℎ, for any g, ℎ ∈ G and x ∈ R. Thus, for x ∈ Dg−1 we
have
tr�(�g(x)) =
∑
ℎ∈G
�ℎ
(
�g
(
x1g−1
)
1ℎ−1
)
=
∑
ℎ∈G
�ℎg
(
x1g−1ℎ−1
)
1ℎ
=
∑
u∈G
�u (x1u−1) 1u1ug−1 =
∑
u∈G
�u (x1u−1)�u
(
1u−11g−1
)
=
∑
u∈G
�u
(
x1g−11u−1
)
=
∑
u∈G
�u (x1u−1) = tr�(x).
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.52 Morita context and partial actions of groups on rings
1.2. Proposition. The applications Γ and Γ′, defined in (1) and (2) are
well defined and are respectively (R�, R�)-bimodule and (S, S)-bimodule
homomorphisms.
Proof. Consider Γ̄ : V ×W → R�, defined by Γ̄(x, y) = tr�(xy), for all
x, y ∈ R. We will prove that Γ̄ is S-balanced, hence Γ is well defined.
Actually, let r ∈ V , r′ ∈ W , g ∈ G and a ∈ Dg. Since r (a�g ⋅ r
′) =
ra�g(r
′1g−1) = �g
(
�g−1 (ra) r′
)
and (r ⋅ a�g) r
′ = �g−1 (ra) r′ ∈ Dg−1 ,
then by Lemma 1.1, we have that Γ̄(r, a�g ⋅ r
′) = tr�(�g
(
�g−1 (ra) r′
)
) =
tr�
(
�g−1 (ra) r′
)
= Γ̄(r⋅a�g, r
′). The other properties of Γ are immediate.
In a similar way we will check that Γ′ is well defined considering Γ̄′ :
W × V → S, defined by Γ̄′(x, y) =
∑
g∈G
x�g(y1g−1)�g, for all x, y ∈ R.
For t ∈ R� and r, r′ ∈ R it easily follows that Γ̄′(rt, r′) = Γ̄′(r, tr′).
Further Γ′ is an (S, S)-bimodule homomorphism. Actually, for all ℎ ∈ G
and y ∈ R, we have
∑
g∈G
�ℎ
(
�g(y1g−1)1ℎ−1
)
�ℎg =
∑
g∈G
�ℎg(y1(ℎg)−1)�ℎg =
∑
u∈G
�u(y1u−1)�u. Therefore
a�ℎΓ
′ (x⊗ y) =
∑
g∈G
a�ℎ
(
x�g(y1g−1)1ℎ−1
)
�ℎg
=
∑
g∈G
a�ℎ(x1ℎ−1)�ℎ
(
�g(y1g−1)1ℎ−1
)
�ℎg
=
∑
u∈G
(a�ℎ ⋅ x)�u(y1u−1)�u = Γ′ ((a�ℎ ⋅ x)⊗ y) .
Finally,
Γ′ (x⊗ y) (a�ℎ) =
∑
g∈G
(
x�g(y1g−1)�g
)
(a�ℎ) =
∑
g∈G
x�g
(
ya1g−1
)
�gℎ
=
∑
u∈G
x�uℎ−1 (ya1ℎu−1) �u =
∑
u∈G
x�u (�ℎ−1 (ya) 1u−1) �u
= Γ′ (x⊗ �ℎ−1 (ya)) = Γ′ (x⊗ (y ⋅ a�ℎ)) .
Thus, the proposition is proved.
1.3. Remark. From Proposition 1.2, for any x, y ∈ R, we get one sided
ideals Γ(V ⊗S x) < R�R�, Γ(y ⊗S W ) < R�
R� , Γ′(x ⊗R� V ) < SS and
Γ′(W ⊗R� y) < SS. In particular, Γ(V ⊗S W ) is an ideal of R� and
Γ′(W ⊗R� V ) is an ideal of S.
It remains to verify the associativity conditions.
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.J. Guzmán, J. Lazzarin 53
1.4. Proposition. Using the previous notations, we have x ⋅Γ′(y⊗ z) =
Γ (x⊗ y) ⋅ z and Γ′(x⊗ y) ⋅ z = x ⋅ Γ (y ⊗ z) for all x, y, z ∈ R.
Proof. Let x, y, z ∈ R. Then x ⋅ Γ′(y ⊗ z) = x ⋅
∑
g∈G
y�g(z1g−1)�g =
∑
g∈G
�g−1
(
xy�g(z1g−1)
)
=
∑
g∈G
�g−1 (xy1g) z = tr�(xy)z = Γ (x⊗ y) ⋅ z.
Moreover,
Γ′(x⊗ y) ⋅ z =
∑
g∈G
x�g(y1g−1)�g ⋅ z =
∑
g∈G
x�g
(
yz1g−1
)
= xtr�(yz) = x ⋅ Γ (y ⊗ z) .
Thus, the assertions hold.
As an immediate consequence of Propositions 1.2 and 1.4 we obtain
1.5. Theorem. Using the previous notations, the six-tuple (R�, S =
R ★� G, V,W,Γ,Γ′) is a Morita context.
As simple application of Theorem 20 and Corollary 23 of [1] we have
1.6. Corollary. Using the previous notations, the following assertions
hold:
1. Γ(V ⊗S rad(S)W ) ⊆ rad (R�).
2. Γ′(W ⊗R� rad(R�)V ) ⊆ rad (S) .
In both cases, rad denotes one of the following radicals: Prime, Jacobson,
Levitzki or the Nil upper if R satisfies Köthe’s Conjecture.
We will keep throughout all the next sections the same notations
introduced in this one.
2. Non-degeneracy of Γ and Γ
′
Recall that, if A,B and C are additive groups, a bilinear form F : A×B →
C is nondegenerate if, for all 0 ∕= a ∈ A and 0 ∕= b ∈ B, we have
F (a,B) ∕= 0 and F (A, b) ∕= 0. The non-degeneracy of Γ and Γ′ provides
some consequences that we will list in this section. Firstly, we also recall
that an ideal I of R is said to be �-invariant if �g(I ∩Dg−1) ⊆ I ∩Dg, for
any g ∈ G. Note that this notion is equivalent to �g(I ∩Dg−1) = I ∩Dg,
for any g ∈ G, (see [6], Definition 2.1).
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.54 Morita context and partial actions of groups on rings
2.1. Lemma. If x ∈ W , then x⊥ = {y ∈ V : Γ′(x⊗ y) = 0} is a right
�-invariant ideal of R contained in ranR(x) (the right annihilator of x
in R). Analogously, if y ∈ V , then y⊥ = {x ∈ W : Γ′(x⊗ y) = 0} is a
left �-invariant ideal of R contained in lanR(y) (the left annihilator of y
in R).
Proof. Consider x ∈ W, r ∈ R and y ∈ x⊥, we have Γ′(x ⊗ yr) =
Γ′(x ⊗ y ⋅ r�1) = Γ′(x ⊗ y)r�1 = 0r�1 = 0, thus yr ∈ x⊥. Now 0 =
Γ′(x ⊗ y) = x
∑
g∈G �g(y1g−1)�g and hence 0 = x�1(y1R)�1 = xy im-
plies y ∈ ranR(x). It follows that x⊥ ⊆ ranR(x). Moreover, since
Γ′(x ⊗ �g(y1g−1)) = Γ′(x ⊗ y ⋅ 1g−1�g−1) = Γ′(x ⊗ y) ⋅ 1g−1�g−1 = 0,
it follows that �g(y1g−1) ∈ x⊥. The second assertion follows by similar
arguments.
2.2. Lemma. Γ is nondegenerate if and only if Γ′ is nondegenerate.
Proof. Let r ∈ R. By Proposition 1.4 we have RΓ(r⊗S W ) = Γ′(W ⊗R�
r)R and RΓ′(r ⊗R� V ) = Γ(V ⊗S r)R. Since R is unital, the result
follows.
In the following proposition we will see that Γ and Γ′ are nondegener-
ate and, as a consequence, that some radical properties are transferable
from the partial skew group ring to the partial fixed ring.
2.3. Proposition. The following statements hold:
1. Γ and Γ′ are nondegenerate.
2. rad (S) = 0 if and only if rad (R�) = 0, where rad denotes someone
of the following radicals: Prime, Jacobson, Levitzki or the Nil upper
radical if R satisfies the Köthe’s conjecture.
3. If I < SS is minimal, then V ⋅I = (0) or V ⋅I is a simple R�-module
Proof. 1. Take x ∈ R, x ∕= 0. Since R is unital, we have that ranR (x) ∕=
R. Now using the Lemma 2.1, x⊥ ⊆ ranR (x) ∕= R implies that there
exists y ∈ V such that Γ′(x ⊗ y) ∕= 0. Hence Γ′(x ⊗R� V ) ∕= 0. In an
analogous way, we get that lanR (y) ∕= R and Γ′(W ⊗R� y) ∕= 0 for any
0 ∕= y ∈ V .
2. The result follows from item 1 and Corollary 1.6.
3. Assume that V ⋅ I ∕= 0 and consider 0 ∕= J ⊆ V ⋅ I, where J is
a left R�-submodule of R. By item 1, Γ′(W ⊗R� J) ∕= 0. Then 0 ∕=
Γ′(W ⊗R� J) ⊆ Γ′(W ⊗R� V ⋅I) = Γ′(W ⊗R� V )I ⊆ I. Since I is minimal
in S, it follows that Γ′(W ⊗R� J) = I, hence V ⋅ Γ′(W ⊗R� J) = V ⋅ I.
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.J. Guzmán, J. Lazzarin 55
Now, by Proposition 1.4, we have V ⋅ Γ′(W ⊗R� J) = Γ(V ⊗S W )J ⊆ J .
Therefore, J ⊆ V ⋅ I = V ⋅Γ′(W ⊗R� J) ⊆ J , that is, J = V ⋅ I, thus V ⋅ I
is a simple R�-module.
Recall that N is an essential submodule of a module M if, for all
nonzero submodules X of M , one has N ∩X ∕= 0. If an ideal (resp.a left
ideal, a right ideal) I is an essential submodule of RRR (resp. RR, RR)
it is called an essential (resp. left, right) ideal.
2.4. Proposition. The following statements hold:
1. If x ∈ R is such that Γ′(W ⊗R� V ) ⋅x = 0, then x = 0; analogously,
if y ∈ R, is such that y ⋅ Γ′(W ⊗R� V ) = 0, then y = 0.
2. lanR (Γ(V ⊗S W )) = ranR (Γ(V ⊗S W )) = 0. In particular,
Γ(V ⊗S W ) is an essential ideal of R�.
3. If A is a subset of R� and lanR� (A) = 0, then lanR (A) = 0. The
same holds for right annihilators.
4. If E is an essential submodule of RR or R�R, then Γ(V ⊗S E) is
an essential submodule of R�R�.
Proof. 1. It is an immediate consequence of Propositions 1.4 and 2.3.
2. By Proposition 2.3, we have that Γ′ is nondegenerate, then we can
prove that lanR (Γ(V ⊗S W )) = ranR (Γ(V ⊗S W )) = 0 using similar
arguments as in 1. So, it follows that Γ(V ⊗S W ) is an essential ideal
of R�.
3. For A ⊆ R�, Γ(1R ⊗S lanR(A)) ⊆ lanR�(A). Actually Γ(1R ⊗S
lanR(A)) ⊆ tr�(R) ⊆ R� and Γ(1R⊗S lanR(A))A = tr�(lanR(A)A) = 0.
Again, since Γ(ranR(A)⊗SW ) ⊆ tr�(R) ⊆ R� and AΓ(ranR(A)⊗SW ) =
Γ(AranR(A)⊗S W ) = 0, it follows that Γ(ranR(A)⊗S W ) ⊆ ranR� (A).
By the non-degeneracy of Γ, we obtain the result.
4. Let E be a essential left ideal of R and 0 ∕= J < R�R�. Hence
0 ∕= J ⊆ RJ < RR implies RJ ∩ E ∕= 0. Thus there exist n > 0,
r1, ⋅ ⋅ ⋅ , rn ∈ R and j1, ⋅ ⋅ ⋅ , jn ∈ J , such that 0 ∕=
n
∑
i=1
riji ∈ E. By
assumption, we have 0 ∕= Γ(V ⊗S
n
∑
i=1
riji) =
n
∑
i=1
Γ(V ⊗S ri)ji ⊆ J . Hence
Γ(V ⊗S E) ∩ J ∕= 0. The remaining part follows similarly.
Following [10], Chapter 1, we say that � has a nondegenerate partial
trace if R� is semiprime and for any non-zero left �-invariant ideal H of
R we have tr�(H) ∕= 0. It is easy to see that if R� is semiprime and Γ is
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.56 Morita context and partial actions of groups on rings
nondegenerate then � has a nondegenerate partial trace. We will use this
in the next result. Before, recall that a nonzero left module U is uniform
if each nonzero left submodule of U is essential in U . We also recall that
a left module M is said to have finite uniform dimension if it contains no
infinite direct sum of nonzero left submodules. In this case, any direct
sum of uniform left submodules of M which is essential in M has precisely
the same quantity of summands. Such quantity is called the left uniform
dimension of M , and is written udimM . In particular, if R is a ring,
udimR will denote the left uniform dimension of RR. Finally, a ring R
is a left Goldie ring if it has finite left uniform dimension and satisfies
the ascending chain condition on the left annihilators (see [9], Sections
2.2 and 2.3, for details). Theorems 5.5 and 5.6 of [7] assert that if R is a
semiprime ∣G∣-torsion free ring, then udimR� ≤ udimR ≤ ∣G∣udimR�.
This same result also holds under another different hypotheses.
2.5. Corollary. Assume that R and R� are semiprime. If R is a left
Goldie ring, then R� is a left Goldie ring. Furthermore udimR� ≤
udimR ≤ ∣G∣udimR�.
Proof. By Proposition 2.3, the first part is immediate from Corollary 5.2
of [7]. Now, by Corollary 1.15 of [6] and Theorem 1.4 of [7] we have
that the enveloping T and its subring TG are semiprime. Then � and
its enveloping action have a nondegenerate partial trace on R and T
respectively. By the first part of this corollary applied to T , Proposition
1.18 of [6], Theorem 1.4 of [7] and Theorem 5.3 of [10], the result follows.
We finish this section with an example showing that the hypotheses
of Corollary 2.5 are, in fact, not equivalent to the claimed in Theorems
5.5 and 5.6 of [7].
2.6. Example. Take R = Ke1 ⊕ Ke2 ⊕ Ke3, where K is a ring and
e1, e2, e3 are orthogonal central idempotents of R. Let G be the cyclic
group of order 5 with generator g and define a partial action of G on R by:
�1 = idR, �g : Ke1 ⊕Ke2 → Ke2 ⊕Ke3, �g(e1) = e2 and �g(e2) = e3;
�g2 : Ke1 → Ke3, �g2(e1) = e3; �g3 : Ke3 → Ke1, �g3(e3) = e1;
�g4 : Ke2 ⊕ Ke3 → Ke1 ⊕ Ke2, �g4(e2) = e1 and �g4(e3) = e2. If
K = ℤ/15ℤ we have that R� = K1R. Then R and R� are semiprime
rings, but R is not a ∣G∣-torsion free.
3. Morita equivalence
The main purpose of this section is to show that the existence of partial
Galois coordinates of R over R� is a necessary and sufficient condition
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.J. Guzmán, J. Lazzarin 57
for the map Γ′ to be surjective, and if in addition the trace map tr� from
R to R� is onto then the Morita context (R�, S = R ★� G, V,W,Γ,Γ′) is
strict.
Recall from [5] Section 3, that R is an �-partial Galois extension of
R� if there exist elements xi, yi ∈ R, i = 1, ..., n, such that
n
∑
i=1
xi�g
(
yi1g−1
)
= �1,g1R,
for any g ∈ G. Such elements are called partial Galois coordinates of R
over R�.
3.1. Theorem. The following statements are equivalent:
1. R is an �-partial Galois extension of R�.
2. R is a finitely generated projective right R�-module and
' : S −→ End(RR�) defined by '(a�g)(x) = a�g(x1g−1) is an
isomorphism of rings.
3. RtR = S, where t =
∑
ℎ∈G 1ℎ�ℎ.
4. The map Γ′ is surjective.
5. R is a generator for the category of the left S-modules.
Moreover, if at least one of the above statements holds, then the following
additional statements also are equivalent:
6. R� = tr�(R).
7. R is a generator for the category of the right R�-modules.
8. The Morita context (R�, S = R ★� G, V,W,Γ,Γ′) is strict.
Proof. 1. ⇔ 2. It follows by the same arguments used in the proof of
Theorem 4.1 of [5].
1. ⇔ 3. It suffices to observe that RtR = S if and only if there exists
elements a1, ..., an, b1, ..., bn ∈ R such that
∑n
i=1 aitbi = 1R, if only if
{ai, bi}
n
i=1 are partial Galois coordinates.
1. ⇔ 4. Γ′ is onto if and only if there exist elements xi, yi ∈ R,
1 ≤ i ≤ n such that
n
∑
i=1
∑
g∈G xi�g
(
yi1g−1
)
�g = 1, if and only if there
exist elements xi, yi ∈ R, 1 ≤ i ≤ n such that
n
∑
i=1
xi�g
(
yi1g−1
)
= �1,g1R,
for any g ∈ G.
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.58 Morita context and partial actions of groups on rings
2. ⇔ 5. We have that (R�)op ⋍ EndS (R). Actually, For all a ∈ R�
we define � : R� → EndS (R) by �(a) = �a where �a(x) = xa, for all
x ∈ R. Since �a (u�g ⋅ r) = u�g(r1g−1)a = u�g(ra1g−1) = u�g ⋅ (r�a),
for any u ∈ Dg and r ∈ R, it follows that �a ∈ EndS (R). It is easy to
see that � is a monomorphism of rings. Let f ∈ EndS (R) and r ∈ R.
Since f(r) = f (r�1 ⋅ 1R) = r�1 ⋅ f (1R) = rf(1R) and for any g ∈ G,
�g(f(1R)1g−1) = 1g�g ⋅ f(1R) = f(1g�g ⋅ 1R) = f(1g) = 1gf(1R), we have
that � is an isomorphism. Finally, from Theorem 0.4 of [10] we have the
equivalence.
Now by assumption that one of the above statements holds,
6. ⇔ 7. Assuming that R� = tr�(R) it follows that the map tr� is
surjective and so R is a right R�-generator. Conversely, first observe that
R is a right R�-generator if and only if the trace ideal of R�, defined by
T (RR�) :=
∑
f∈Hom(RR� ,R�) f(R), equals R� (see, for instance Theorem
18.8 of [8]). Now, take f ∈ Hom(RR� , R�). By the assertion 3. above,
there exists y ∈ S, y =
∑
g∈G ag�g, such '(y) = f . Then for any r ∈ R
we have '(y)(r) =
∑
g∈G ag�g(r1g−1) ∈ R�. Thus, for any ℎ ∈ G,
∑
g∈G
ag�g(r1g−1)1ℎ = �ℎ(
∑
g∈G
ag�g(r1g−1)1ℎ−1)
=
∑
g∈G
�ℎ(ag1ℎ−1)�ℎ
(
�g(r1g−1)1ℎ−1
)
=
∑
g∈G
�ℎ (ag1ℎ−1)�ℎg
(
r1(ℎg)−1
)
=
∑
�∈G
�ℎ (aℎ−1�1ℎ−1)�� (r1�−1) .
and so
'(
∑
g∈G
ag1ℎ�g)(r) = '(
∑
g∈G
�ℎ
(
aℎ−1g1ℎ−1
)
�g)(r).
Hence
∑
g∈G
ag1ℎ�g =
∑
g∈G
�ℎ
(
aℎ−1g1ℎ−1
)
�g,
which implies
ag1ℎ = �ℎ
(
aℎ−1g1ℎ−1
)
.
for any g, ℎ ∈ G. In particular, for ℎ = g, we have ag = �g(a11g−1).
Therefore, y =
∑
g∈G ag�g =
∑
g∈G �g(a11g−1)�g = ta1, f = tr�(a1_)
and, consequently, R� ⊆ tr�(R).
7. ⇔ 8. Immediate.
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.J. Guzmán, J. Lazzarin 59
3.2. Corollary. Suppose that at least one of the elements tr�(1R) and
∣G∣1R is invertible in R. Then, R is an �-partial Galois extension of R�
if and only if the Morita context (R�, S = R ★� G, V,W,Γ,Γ′) is strict.
Proof. It suffices to show that tr�(R) = R�. One easily sees that tr�(1R)
is invertible in R if and only if there exists c ∈ R� such that tr�(c) = 1R.
And if ∣G∣1R is invertible in R then the result follows from Lema 2.1 of
[5] and Proposition 2.5 of [2].
4. Applications
The main purpose of this section is to establish some sufficient conditions
for a ring R to be an �-partial Galois extension of R�. Recall that a right
(resp. left) S-module M is faithful if annMS = 0 (resp. ann SM = 0). In
general, V and W are not faithful S-modules (see Example 2.1 of [3]). In
fact, it easily follows from Proposition 1.2 and the non-degeneracy of Γ′
that annVS = ranSΓ
′ (W ⊗R� V ) and ann SW = lanSΓ
′ (W ⊗R� V )).
4.1. Proposition. The following statements hold:
1. If VS is faithful, then Γ′(W ⊗R� V ) is a essential left ideal of S.
2. If SW is faithful, then Γ′(W ⊗R� V ) is a essential right ideal of S.
Moreover, if R is semiprime and ∣G∣-torsion free, then the converse
of 1. and 2. also holds.
Proof. 1. If VS is faithful, then we have ranSΓ
′ (W ⊗R� V ) = 0. Now,
consider a left ideal J of S such that J ∩ Γ′ (W ⊗R� V ) = 0. Then we
have Γ′ (W ⊗R� V ) J ⊆ J ∩ Γ′ (W ⊗R� V ) = 0 and so
J ⊆ ranSΓ
′ (W ⊗R� V ) = 0.
2. It is analogous to item 1.
Finally assume that R is semiprime and ∣G∣-torsion free. Then by
Proposition 5.3 of [6], we have that S is a semiprime ring. Therefore, if
Γ′(W ⊗R� V ) is a essential right ideal of S, we have ranSΓ
′ (W ⊗R� V ) =
0 and so VS is faithful. The converse of the item 2. follows similarly.
4.2. Remark. We observe that if R is semisimple and tr�(1R) is invert-
ible in R, then the assertions 1. and 2. in Proposition 4.1 are in fact
equivalences, by Corollary 6.8 of [6].
We end with the following proposition which gives some sufficient
conditions on R in order to obtain a Morita equivalence between R�
and S.
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.60 Morita context and partial actions of groups on rings
4.3. Proposition. Assume that R is a ring such that RS (or SR) is
faithful. If R is semisimple and at least one of the elements tr�(1R) or
∣G∣1R is invertible in R, then R is an �-partial Galois extension of R�.
In particular, R� and S are Morita equivalents.
Proof. By Maschke Theorem (see Theorem 3.1 or Corollary 3.3 of [6] for
the partial case) we have that S is semisimple. Now, by Proposition 4.1,
Γ′(V ⊗R� W ) is a essential left (or right) ideal of S. Thus Γ′(V ⊗R� W ) =
S, that is, Γ′ is onto. Then, the result follows by Corollary 3.2.
Acknowledgments
The authors are grateful to the referee for several comments which help
them to improve the first version of this paper.
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Contact information
Jesús Ávila G. Departamento de Matemáticas y Es-
tad́ıstica, Universidad del Tolima, Ibagué,
Colombia.
E-Mail: javila@ut.edu.co
URL: http://www.ut.edu.co
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.J. Guzmán, J. Lazzarin 61
João Lazzarin Departamento de Matemática, Universi-
dade Federal de Santa Maria, 97105-900,
Santa Maria-RS, Brazil.
E-Mail: lazzarin@smail.ufsm.br
URL: http://w3.ufsm.br/depmat/
Received by the editors: 30.08.2009
and in final form 20.09.2009.
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