GLOBALIZATIONS OF PARTIAL ACTIONS ON NON UNITAL RINGS
MICHAEL DOKUCHAEV, ÁNGEL DEL RÍO, AND JUAN JACOBO SIMÓN
Abstract. In this note we prove a criteria for the existence of a globalization for a given
partial action of a group on an s-unital ring. If the globalization exists, it is unique in a
natural sense. This extends the globalization theorem from [6] obtained in the context
of rings with 1.
1. Introduction
Partial actions of groups in a systematic way appeared in the theory of operator algebras
in a process of a generalization of C ∗ -crossed products, and the main steps were made
in [8], [16] and [9]. Since then important classes of C ∗ -algebras have been described as
crossed products by partial actions, including the case of Cuntz-Krieger algebras [5], deeply
investigated in [11] and [12] from the point of view of partial representations, a related
concept which also appeared in the theory of operator algebras. The algebraic study of
partial actions of groups on abstract algebras and the corresponding crossed products was
begun in [6]. It was shown, in particular, that an essential part of elementary gradings
on matrix algebras come from generalized crossed product structures. Further algebraic
results involving partial actions on rings were obtained in [4], [7] and [13].
It seems that when a partial action on some structure is given, one of the most relevant
problems is the question of the existence and uniqueness of a globalization, also called
an enveloping action, i.e. of a global action, whose restriction to the original object gives
the initial partial action. The study of this problem was initiated independently in [1]
and [15]. Inspired by [15], further results on globalizations were obtained in [18]. In [6] a
criteria of the existence of a globalization for a partial action on a unital ring was given.
However, important examples of partial actions on rings occur in the non unital context.
For instance, by Gelfand’s Theorem, any commutative C ∗ -algebra is of the form C0 (X), the
algebra of the complex valued continuous functions on a locally compact Hausdorff space
X which vanish at infinity. It is shown in [2] that the partial actions of a locally compact
Hausdorff group G on X naturally correspond to the partial actions of G on C0 (X).
However, if X is not compact then C0 (X) is non unital. Furthermore, globalizations of
partial actions on C ∗ -algebras not always exist (see [1]). Thus it is rather desirable to
examine large classes of non unital rings with respect to the globalization problem.
If a partial action on a unital ring has a globalization then it is unique in a canonical
way [6] and the ring on which this globalization is defined is s-unital but not necessarily unital. Observe that in the theory of operator algebras there are highly important
examples of s-unital rings. One is given by the algebra Cc (X) of the complex valued continuous functions on X with compact supports, where X is a non compact locally compact
Research supported by Capes of Brazil, D.G.I. of Spain and Fundación Séneca of Murcia.
1
2
M. DOKUCHAEV, A. DEL RÍO, AND J. J. SIMÓN
Haudorff space. Note that Cc (X) has no idempotents and it is dense in C0 (X). Another
example is the algebra of the finite-rank operators on a Hilbert space H. It is dense in
the C ∗ -algebra of the compact operators on H. Thus it is natural to treat our problem in
the s-unital context. This is the purpose of this paper in which we obtain a criteria to
decide if a partial action on an s-unital ring has a globalization. We also prove that when
a globalization of a partial action on an s-unital ring exists, it is unique and, moreover,
the partial crossed product and the global one are Morita equivalent.
2. Preliminaries
The first formal definition of a partial action was given by R. Exel in [10]. Thus while a
global action of a group G on a set X is a collection of bijections X → X which agree with
the group structure, a partial action of G on X is a family of bijections αg : Xg−1 → Xg
between subsets of X such that αg ◦ αh is a restriction of αgh . Usually one also requires
that the identity element of G acts as the trivial bijection X → X . When X has some
additional structure, one has to specify this definition, we do it below for rings. By a ring
we shall mean an associative non-necessarily unital ring.
Definition 2.1. Let G be a group with identity element 1 and A be a ring. A partial action
α of G on A is a collection of (two-sided) ideals Ag ⊆ A (g ∈ G) and ring isomorphisms
αg : Ag−1 → Ag such that
(i) A1 = A and α1 is the identity map of A;
(ii) A(gh)−1 ⊇ α−1
h (Ah ∩ Ag −1 );
(iii) αg ◦ αh (x) = αgh (x) for each x ∈ α−1
h (Ah ∩ Ag −1 ).
The conditions (ii) and (iii) mean that the function αgh is an extension of the function
αg ◦ αh . Moreover, it is easily seen that (ii) can be replaced by a “stronger looking” condition:
(ii’) αg (Ag−1 ∩ Ah ) = Ag ∩ Agh , for all g, h ∈ G,
(see [6]).
Examples of partial actions can be obtained by restricting (global) actions to nonnecessarily invariant (two-sided) ideals. Indeed, suppose that a group G acts on a ring
B by automorphisms βg : B → B and let A be an ideal of B. Set Ag = A ∩ βg (A) and
let αg be the restriction of βg to Ag−1 . It is easily verified that we have a partial action
α = {αg : Ag−1 → Ag , g ∈ G} of G on A. One of the general problems is to determine
the conditions under which a given partial action can be obtained as the restriction of
a (global) action, and analyse the uniqueness of such a globalization up to isomorphism.
This problem was resolved in [6, Theorem 4.5] for partial actions on unital rings. Next we
recall the precise meaning of a globalization.
GLOBALIZATIONS OF PARTIAL ACTIONS ON NON UNITAL RINGS
3
Definition 2.2. An action β of a group G on a ring B is said to be a globalization (or an
enveloping action) for the partial action α of G on a ring A if there exists a monomorphism
ϕ : A → B such that:
(i) ϕ(A) P
is an ideal in B,
(ii) B = g∈G βg (ϕ(A)),
(iii) ϕ(Ag ) = ϕ(A) ∩ βg (ϕ(A)),
(iv) ϕ ◦ αg = βg ◦ ϕ on Ag−1 .
Definition 2.3. We say that the globalization (β, B) of a partial action (α, A) is unique
if for any other globalization (β 0 , B 0 ) of (α, A) there exists an isomorphism of rings
ϕ : A → A0 such that
β 0 g ◦ ϕ = ϕ ◦ βg ,
for every g ∈ G.
Given a partial action α of a group G on a ring A,
P the skew group ring A ∗α G corresponding to α is the set of all finite formal sums { g∈G ag ug : ag ∈ Ag }, where ug are
symbols. The addition is defined in the obvious way, and the multiplication is determined
by (ag ug ) · (bh uh ) = αg (αg−1 (ag )bh )ugh . Obviously, A 3 a 7→ au1 ∈ A ∗α G is an embedding which permits us to identify A with Au1 . The first question which naturally arises is
whether or not A ∗α G is associative. It is shown in [6] that this is not always the case,
however, if A is semiprime, A ∗α G is necessarily associative (see [6, Corollary 3.4]).
If (α, A) admits a globalization (β, B), then the skew group ring A ∗α G has an embedding into B ∗β G. In particular, A ∗α G is associative.
We shall need some easy but important for our purpose results on s-unital rings. We
recall that a left s-unital ring, by definition, is an associative ring A such that for any
x ∈ A one has x ∈ Ax. We say that e ∈ A is a left unit for x ∈ A if ex = x. Of course every
unital ring is s-unital. Other examples of s-unital rings are rings with local units (see e.g.
[19]). It is interesting to observe that any C ∗ -algebra A is s-unital in an approximate
sense: for each a ∈ A and arbitrary ε > 0 there exists e ∈ A with ||a − ea|| < ε (see, for
example, [17, Theorem 3.1.1]).
Lemma 2.4. Suppose that the ideals I and J of a ring A are left s-unital rings. Then
I + J and I ∩ J are also left s-unital and, moreover, I ∩ J = IJ . Furthermore, for
arbitrary a, b ∈ I there exists e ∈ I such that ea = a and eb = b.
Proof. Take a ∈ I, b ∈ J and suppose that x ∈ I is a left unit for a and y ∈ J is a left
unit for xb − b ∈ J . Then it is easily seen that x + y − yx is a left unit for a + b. Take now
a ∈ I ∩ J and let x ∈ I, y ∈ J be left units for a. Then obviously z = xy ∈ I ∩ J is a
left unit for a. Moreover, I ∩ J 3 a = za ∈ IJ implies I ∩ J ⊆ IJ . Finally, for arbitrary
a, b ∈ I let x, y ∈ I be such that xa = a and y(xb − b) = xb − b. Then it is easy to see
that taking e = x + y − yx one has ea = a and eb = b.
Remark 2.5. It follows by Lemma 2.4 that a (non-necessarily finite) sum of left s-unital
ideals in a ring A is left s-unital. For it is enough to observe that any element belongs
to a finite sum of left s-unital ideals. In particular, if (β, B) is a globalization of a partial
action of an s-unital ring A then B is left s-unital too.
4
M. DOKUCHAEV, A. DEL RÍO, AND J. J. SIMÓN
For homomorphisms of left A-modules we shall use the right hand side notation, i. e.
we write x 7→ xγ for γ :A M →A N, while for γ : MA → NA , we use the usual notation:
x 7→ γx. Accordingly, composition of left module homomorphisms is read from left to right
i. e. x(γ1 γ2 ) = (xγ1 )γ2 , while composition of right module homomorphisms is read in the
usual right to left way. We remind that given a ring A, the multiplier ring M (A) of A is
the set
M (A) = {(R, L) ∈ End(A A) × End(AA ) : (aR)b = a(Lb) for all a, b ∈ A}
with component-wise addition and multiplication (for more details see [3] or [6]). For a
multiplier γ = (R, L) ∈ M (A) and a ∈ A we set aγ = aR and γa = La. Thus one always
has (aγ)b = a(γb) (a, b ∈ A). An element a in a ring A obviously determines the multiplier (Ra , La ) ∈ M (A) where xRa = xa and La x = ax (x ∈ A). If I is an ideal in A then
this multiplier evidently restricts to one of I which shall be denoted by the same pair of
symbols (Ra , La ). The first (resp. second) components of the elements of M (A) are called
right (resp. left) multipliers of A.
Lemma 2.6. Let A be a ring.
(i) Suppose that I is an ideal in A and I is s-unital. If φ, ψ : A → I are left A-module
homomorphisms such that φ and ψ coincide on I, then φ = ψ.
(ii) Suppose that A is left s-unital. If γ, γ 0 ∈ M (A) and aγ = aγ 0 for all a ∈ A, then
γ = γ0.
Proof. (i) Given an arbitrary x ∈ A let e ∈ I be a left unit for φ(x) − ψ(x), that is
e(φ(x) − ψ(x)) = φ(x) − ψ(x). Because φ and ψ are left A-module maps, we have φ(x) −
ψ(x) = e(φ(x) − ψ(x)) = φ(ex) − ψ(ex) = 0.
(ii) For arbitrary a, b ∈ A we have a(γ 0 b) = (aγ 0 )b = (aγ)b = a(γb), so that A(γ 0 b−γb) =
0, and because A is left s-unital, it follows that γb = γ 0 b.
Remark 2.7. Let I be an ideal in a left s-unital ring A. If γ ∈ M (A) is such that Iγ ⊆ I
then it is easily seen that γI ⊆ I. In fact, for a ∈ I one has γa = e(γa) = (eγ)a ∈ I,
where e ∈ A is a left unit for γa.
3. The Globalization Theorem
The next result extends the globalization theorem obtained in [6] for partial actions on
rings with unity.
Theorem 3.1. Let α be a partial action of a group G on a left s-unital ring A. Then α
admits a globalization if and only if the following two conditions are satisfied:
(i) Ag is a left s-unital ring for every g ∈ G,
(ii) For each g ∈ G and a ∈ A there exists a multiplier γg (a) ∈ M (A) such that
Aγg (a) ⊆ Ag and γg (a), restricted to Ag as a right multiplier, is αg−1 Ra αg .
Moreover, if a globalization exists, it is unique (in the sense of Definition 2.3) and the
ring under the global action is left s-unital.
Proof. The “only if” part. Suppose that an action β of G on a ring B is a globalization of α. Then A is an ideal in B and Ag = A ∩ βg (A). Since the intersection of two
left s-unital ideals is left s-unital, (i) follows. We see that αg−1 Ra αg maps x ∈ Ag to
GLOBALIZATIONS OF PARTIAL ACTIONS ON NON UNITAL RINGS
5
αg (αg−1 (x)a) = xβg (a) and for any y ∈ A one has yβg (a) ∈ A ∩ βg (A) = Ag , so that we
may take γg (a) = (Rβg (a) , Lβg (a) ) ∈ M (A).
The “if” part. Let F = F(G, M (A)) be the ring of all functions from G to M (A),
that is the Cartesian product of the copies of M (A) indexed by the elements of G. For
convenience the element f (g) will be also denoted by f |g (f ∈ F, g ∈ G).
We define a (global) action β of G on F by the formula:
βg (f )|h = f (g −1 h)
(g, h ∈ G, f ∈ F).
For g ∈ G and a ∈ A the element γg (a) ∈ M (A) given by (ii) is uniquely determined in
view of Lemma 2.6. Hence we can define a map ϕ : A → F by setting
ϕ(a)|g = γg−1 (a), g ∈ G.
Because A is left s-unital, ϕ is injective. Taking a, b ∈ A and x ∈ Ag one has
(xγg−1 (a))γg−1 (b) = ((x)αg−1 Ra αg )αg−1 Rb αg = (x)αg−1 Rab αg = xγg−1 (ab).
Thus by (i) of Lemma 2.6, γg−1 (a)γg−1 (b) and γg−1 (ab) coincide as right multipliers and
(ii) of Lemma 2.6 implies γg−1 (a)γg−1 (b) = γg−1 (ab). It follows that ϕ is multiplicative and
since it is evidently
additive, we conclude that ϕ is a ring monomorphism.
P
Let B = g∈G βg (ϕ(A)). We want to show that the restriction of β to B is a globalization
for α. We denote this restriction by the same symbol β. We need to check that the axioms
(i)-(iv) of Definition 2.2 hold. Axiom (ii) is clear.
We check first axiom (iv) that is
(1)
βg (ϕ(a)) = ϕ(αg (a)),
for any g ∈ G and a ∈ Ag−1 . Let h ∈ G, φ = βg (ϕ(a))|h = ϕ(a)|g−1 h = γh−1 g (a) and
ψ = ϕ(αg (a))|h = γh−1 (αg (a)) seen as right multipliers of A. We have to show φ = ψ.
By Lemma 2.6 it is enough to show that Aφ, Aψ ⊆ I and φ and ψ coincide in I, where
I = Ah−1 ∩ Ah−1 g , an s-unital ideal of A.
Taking x ∈ A let e ∈ Ah−1 g be a left unit for xγh−1 g (a). Then we see that
xφ = xγh−1 g (a) = e(xγh−1 g (a)) = (ex) · γh−1 g (a)
and since ex ∈ Ah−1 g , this equals to
αh−1 g (αg−1 h (ex)a) ∈ αh−1 g (Ag−1 h ∩ Ag−1 ) = I.
Similarly, if f ∈ Ah−1 is a left unit for xγh−1 (αg (a)), we have
xψ = xγh−1 (αg (a)) = f (xγh−1 (αg (a))) = (f x) · γh−1 (αg (a))
= αh−1 (αh (f x)αg (a)) ∈ αh−1 (Ah ∩ Ag ) = I.
This shows that Aφ, Aψ ⊆ I.
For x ∈ I using (iii) of Definition 2.1 one has
xφ = xγh−1 g (a) = αh−1 g (αg−1 h (x)a) = (αh−1 ◦ αg )((αg−1 ◦ αh )(x)a)
= αh−1 (αh (x)αg (a)) = xγh−1 (αg (a)) = xψ.
6
M. DOKUCHAEV, A. DEL RÍO, AND J. J. SIMÓN
This finished the proof of (1). For future use we set the following equality that is exactly
the equality φ = ψ that we just proved:
(2)
γh−1 g (a) = γh−1 (αg (a)),
for any g, h ∈ G, a ∈ Ag−1 .
Next we deal with axiom (iii), that is we show
(3)
ϕ(Ag ) = ϕ(A) ∩ βg (ϕ(A)),
for all g ∈ G. An element from the right hand side can be written as ϕ(a) = βg (ϕ(b))
for some a, b ∈ A. For h ∈ G we see that ϕ(a)|h = γh−1 (a) and βg (ϕ(b))|h = ϕ(b)|g−1 h =
γh−1 g (b). Thus
γh−1 (a) = γh−1 g (b)
and taking h = 1 we obtain a = γg (b) ∈ Ag . This proves the inclusion ϕ(Ag ) ⊇ ϕ(A) ∩
βg (ϕ(A)).
For the converse inclusion take a ∈ Ag and set b = αg−1 (a). Then
ϕ(a)|h = ϕ(αg (b))|h = γh−1 (αg (b)) = γh−1 g (b),
in view of (2). On the other hand, γh−1 g (b) = βg (ϕ(b))|h , by the definition of β. Consequently, ϕ(Ag ) ⊆ ϕ(A) ∩ βg (ϕ(A)) and (3) follows.
To prove that (β, B) is a globalization of (α, A) it only remains to show that ϕ(A) is an
ideal in B. For this we need two properties of γg (a). The first one is:
(4)
Ah γg (a) ⊆ Ag ∩ Ah
for any g ∈ G, a ∈ A. Indeed, for x ∈ Ah let e ∈ Ag be a left unit for xγg (a). Then
xγg (a) = e(xγg (a)) = αg (αg −1 (ex)a) ⊆ Ag ∩ Ah . The second property is:
(5)
Aγg (a) ⊆ Ag ∩ Agh
for all g ∈ G and a ∈ Ah . For take x ∈ A arbitrarily and let e ∈ Ag be a left unit for xγg (a).
Then as above xγg (a) = αg (αg −1 (ex)a) which lies in Ag ∩ Agh as αg −1 (ex)a ∈ Ag−1 ∩ Ah .
We check first that ϕ(A) is a right ideal in B. Since B is the sum of the βg (ϕ(A))’s,
it is enough to show that ϕ(A)βg (ϕ(A)) ⊆ ϕ(A) for all g ∈ G. We do so by proving the
following formula
(6)
ϕ(a)βg (ϕ(b)) = ϕ(aγg (b))
for any a, b ∈ A and g ∈ G. Indeed, taking h ∈ G we set φ = ϕ(a)|h βg (ϕ(b))|h =
ϕ(a)|h ϕ(b)|g−1 h = γh−1 (a)γh−1 g (b) and ψ = ϕ(aγg (b))|h = γh−1 (aγg (b)). We have to show
that φ = ψ. Let I = Ah−1 ∩ Ah−1 g , which is an s-unital ideal of A. Using Lemma 2.6, it
is enough to show that Aφ, Aψ ⊆ I and φ and ψ coincides as right multipliers on I. The
first is a straightforward consequence of (4) and (5). For the second let z ∈ I. Then
zφ = zγh−1 (a)γh−1 g (b) = αh−1 g ( (αg−1 h ◦ αh−1 )(αh (z)a) b)
= (αh−1 ◦ αg )((αg−1 ◦ αh ◦ αh−1 )(αh (z)a)b) = αh−1 (αg (αg−1 (αh (z)a)b))
= αh−1 ((αh (z)a) · γg (b)) = αh−1 (αh (z)(aγg (b))) = zγh−1 (aγg (b)) = zψ.
This shows that ϕ(A) is a right ideal in B.
GLOBALIZATIONS OF PARTIAL ACTIONS ON NON UNITAL RINGS
7
Observe that we did not use so far that γg (a) is a left multiplier. However we need this
to show that ϕ(A) is a left ideal in B. For that we prove the following formula
βg (ϕ(a))ϕ(b) = ϕ(γg (a) · b)
(7)
for any a, b ∈ A and g ∈ G, in which γg (a) acts on b as a left multiplier. Let h ∈ G and
set φ = βg (ϕ(a))|h ϕ(b)|h = γh−1 g (a)γh−1 (b) and ψ = ϕ(γg (a)b)|h = γh−1 (γg (a)b). Again
we have to prove φ = ψ and by (4) and (5) both φ and ψ map A to I = Ah−1 ∩ Ah−1 g , as
right multipliers. Thus in view of Lemma 2.6 it is enough to see that they coincide on I.
Taking arbitrary z ∈ I we have
zφ = zγh−1 g (a)γh−1 (b) = αh−1 ( (αh ◦ αh−1 g )(αg−1 h (z)a) b)
= αh−1 ((αh ◦ αh−1 ◦ αg )((αg−1 ◦ αh )(z)a)b) = αh−1 (αg ((αg−1 ◦ αh )(z)a)b)
= αh−1 ((αh (z) · γg (a))b) = αh−1 (αh (z)(γg (a) · b)) = zγh−1 (γg (a)b) = zψ.
The above shows that β is a globalization of α. It remains to prove the uniqueness of
β. Suppose that (β, B) and (β 0 , B 0 ) are two globalizations of (α, A). By Remark 2.5, both
B and B 0 are s-unital. Let ϕ : A → B and ϕ0 : A → B 0 the corresponding embeddings. It
follows from the definition of a globalization that B 0 is the sum
of the ideals βg0 (ϕ0 (A)), g ∈
P
G. Thus an element of B 0 can be written as a finite sum i βg0 i (ϕ0 (ai )) with gi ∈ G and
ai ∈ A. We want to show that the map φ : B 0 → B given by βg0 (ϕ0 (a)) 7→ βg (ϕ(a)), g ∈
P
G, a ∈ A is well defined. Suppose that si=1 βg0 i (ϕ0 (ai )) = 0 and we need to be sure that
Ps
i=1 βgi (ϕ(ai )) = 0.
P 0 0
0
0
0
For all
P h 0∈ G 0and a 0∈ A we have i βh (ϕ (a))βgi (ϕ (ai )) = 0 and applying βh−1 we
obtain i ϕ (a)βh−1 gi (ϕ (ai )) = 0. Observe that
ϕ0 (a)βh0 −1 gi (ϕ0 (ai )) ∈ ϕ0 (A) ∩ βh0 −1 gi (ϕ0 (A)) = ϕ0 (Ah−1 gi )
and thus ϕ0 (a)βh0 −1 gi (ϕ0 (ai )) = ϕ0 (b0i,a ) with b0i,a ∈ Ah−1 gi . Similarly ϕ(a)βh−1 gi (ϕ(ai )) =
ϕ(bi,a ) with bi,a ∈ Ah−1 gi . Let ei,a ∈ Ah−1 gi be a left unit for b0i,a and bi,a . It exists by
Lemma 2.4. Then
ϕ0 (a) βh0 −1 gi (ϕ0 (ai )) = ϕ0 (ei,a ) ϕ0 (a) βh0 −1 gi (ϕ0 (ai )) = ϕ0 (ei,a a) βh0 −1 gi (ϕ0 (ai )).
Since ei,a a ∈ Ah−1 gi we can write ei,a a = αh−1 gi (ẽi,a ) with ẽi,a ∈ Agi −1 h . Hence
ϕ0 (a) βh0 −1 gi (ϕ0 (ai )) = ϕ0 (αh−1 gi (ẽi,a )) βh0 −1 gi (ϕ0 (ai )) = βh0 −1 gi (ϕ0 (ẽi,a )) βh0 −1 gi (ϕ0 (ai ))
= βh0 −1 gi (ϕ0 (ẽi,a ai )) = ϕ0 (αh−1 gi (ẽi,a ai )).
Consequently,
0=
X
X
ϕ0 (a)βh0 −1 gi (ϕ0 (ai )) = ϕ0 (
αh−1 gi (ẽi,a ai )).
i
i
ϕ0
P
This yields that i αh−1 gi (ẽi,a ai ) = 0, as is a monomorphism. Because ei,a is also a left
unit for bi,a , we similarly have
X
X
0 = ϕ(
αh−1 gi (ẽi,a ai )) =
ϕ(a)βh−1 gi (ϕ(ai )).
i
i
8
M. DOKUCHAEV, A. DEL RÍO, AND J. J. SIMÓN
P
Applying
any h ∈ G, we see
Pβh we come to βh (ϕ(a)) i βgi (ϕ(ai )) = 0. Since this holds forP
that B( i βgi (ϕ(ai ))) = 0. Finally, because B is left s-unital, we obtain i βgi (ϕ(ai )) = 0,
as desired.
Thus φ : B 0 → B is a well-defined homomorphism of rings and by symmetry, βg (ϕ(a)) 7→
0
βg (ϕ0 (a)), g ∈ G, a ∈ A also determines a well-defined map φ0 : B → B 0 . Obviously,
φ0 ◦ φ = φ ◦ φ0 = 1 and, consequently, φ is an isomorphism of rings. It is easily seen that
for all g ∈ G one has βg ◦ φ = φ ◦ βg0 and this yields the uniqueness of the globalization in
the sense of Definition 2.3.
We know by Theorem 4.5 of [6] that a partial action α of a group G on a unital ring A
admits a globalization if and only if each Ag is a unital ring. Thus we have the following:
Corollary 3.2. Suppose that α is a partial action of a group G on a unital ring A such
that Ag is non-unital for some g ∈ G. Then there exists h ∈ G such that either Ah is not
s-unital or for some a ∈ A no element of M (A) extends, as a right multiplier, the left
A-module map Ag 3 x 7→ αg (αg−1 (xa)) ∈ Ag and maps A into Ag (as a right multiplier).
Remark 3.3. Observe that when dealing with Theorem 4.5 of [6] one treats {0} as a unital
ring, thus we do not require 1 6= 0 in our definition of a unital ring.
The next example shows that the second alternative in Corollary 3.2 really occurs. This
implies that condition (ii) of Theorem 3.1 can not be omitted.
Example 3.4. Let A be a ring with 1 and I be an ideal in A which is an s-unital but not a
unital ring. Let G be the cyclic group of order 2 with generator g and let α1 be the identity
map A 7→ A and αg be the identity map I 7→ I. Evidently we have a partial action α of
G on A which is not globalizable by Theorem 4.5 of [6]. Hence by Corollary 3.2 the left
A-module map I 3 x 7→ αg (αg−1 (xa)) = xa ∈ I can not be extended to a multiplier γ of
A such that Aγ ⊆ Ag . For a more concrete example take as A the ring of row and column
finite matrices over a field. This ring evidently has 1 and the subset I of the matrices
with finite number of non-zero entries is an ideal which is both left and right s-unital but
is not unital.
Given a globalizable partial action α of a group G on a unital ring A we know by
Theorem 3.1 that the ring B under the global action is left s-unital. If G is finite then B
has 1 by Lemma 4.4 of [6]. However for an infinite G the ring B is not necessarily unital,
as shown in the next example.
Example 3.5. Let G be an infinite group and A be a ring with 1. Let α1 be the trivial
map A 7→ A and set Ag = 0 for each g 6= 1. Then we evidently have a globalizable partial
action α of G on A (see Remark 3.3). It is easily seen that the ring under the globalized
action in the proof of Theorem 3.1 is B = A(|G|) , the direct product of | G | copies of A,
indexed by the elements of G, on which G acts by the permutation of indices. Clearly, B
is not unital.
4. Morita equivalence
Morita’s fundamental results were extended in [14] to the case of idempotent rings,
which permits us to adapt easily the proof of [6, Theorem 5.4] to the more general case of
GLOBALIZATIONS OF PARTIAL ACTIONS ON NON UNITAL RINGS
9
left s-unital rings. The notion of the Morita context for idempotent rings is the same as
for rings with 1, i. e. a Morita context is a six-tuple (R, R0 , M, M 0 , τ, τ 0 ), where:
(a) R and R0 are rings,
(b) M is an R-R0 -bimodule,
(c) M 0 is an R0 -R-bimodule,
(d) τ : M ⊗R0 M 0 → R is a bimodule map,
(e) τ 0 : M 0 ⊗R M → R0 is a bimodule map,
such that
τ (x ⊗ x0 ) y = x τ 0 (x0 ⊗ y),
∀x, y ∈ M, x0 ∈ M 0 ,
and
τ 0 (x0 ⊗ x) y 0 = x0 τ (x ⊗ y 0 ),
∀x0 , y 0 ∈ M 0 , x ∈ M.
Given a Morita context with τ and τ 0 onto, the categories of R-modules and of R0 modules are equivalent, that is R and R0 are Morita equivalent.
We have the following:
Theorem 4.1. Let α be a partial action of a group G on a left s-unital ring A and
suppose that (β, B) is an enveloping action for (α, A). Then A ∗α G and B ∗β G are Morita
equivalent.
Proof. For the sake of completeness we recall the notation from [6]. Set R = A ∗α G and
R0 = B ∗β G and consider the linear subspaces M, N ⊆ B ∗β G given by




X

X

M=
cg ug : cg ∈ A
and N =
cg ug : cg ∈ βg (A) .




g∈G
g∈G
We view A ∗α G as a subring of B ∗β G. The proofs of Propositions 5.1 and 5.2 of [6] do not
use the fact that the ring has 1 and hold for arbitrary (associative) rings. According to
these Propositions M is an A ∗α G-B ∗β G - bimodule and N is a B ∗β G-A ∗α G - bimodule.
Also it is readily verified that M N ⊆ A ∗α G. One defines τ and τ 0 as follows:
τ : m ⊗ n ∈ M ⊗B∗β G N 7−→ mn ∈ A ∗α G
and
τ 0 : n ⊗ m ∈ N ⊗A∗α G M 7−→ nm ∈ B ∗β G.
For arbitrary g ∈ G and a ∈ Ag let e ∈ Ag be a left unit for a. We obviously have
eu1 · aug = aug ,
and since a ∈ αg (Ag−1 ) ⊆ βg (A), the element aug belongs to N. Evidently, eu1 ∈ M and
we have that A ∗α G ⊆ M N, where M N stands for the linear span of the set of products
xy with x ∈ M and y ∈ N . It follows that τ is onto.
Let g, h ∈ G and a ∈ A be arbitrary and let e ∈ A be a left unit for a. Then βg (e)ug ∈ N ,
aug−1 h ∈ M and
βg (e)ug · aug−1 h = βg (ea)uh = βg (a)uh .
10
M. DOKUCHAEV, A. DEL RÍO, AND J. J. SIMÓN
P
Since B = g∈G βg (A) we conclude that B uh ⊆ N M and hence N M = B ∗β G, which
shows that τ 0 is also onto. Thus the six-tuple (A ∗α G, B ∗β G, M, N, τ, τ 0 ) is a Morita
context with τ and τ 0 onto.
Acknowledgments
This work was done during the visit of the first author to the University of Murcia. He
thanks the Department of Mathematics of the University for its warm hospitality.
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Departamento de Matemática, Universidade de São Paulo, Brasil
E-mail address: [email protected]
Departamento de Matemáticas, Universidad de Murcia, España
E-mail address: [email protected]
Departamento de Matemáticas, Universidad de Murcia, España
E-mail address: [email protected]