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Abstract
In this paper, we investigate primeness of groupoid graded rings. We provide a set of necessary and sufficient conditions for primeness of a nearly-epsilon strongly groupoid graded ring. Furthermore, we apply our main result to get a characterization of prime partial skew groupoid rings, and in particular of prime groupoid rings, thereby generalizing a classical result by Connell and partially generalizing recent results by Steinberg.
1 Introduction
Throughout this paper, all rings are assumed to be associative but not necessarily unital. Recall that a ring S is said to be prime if there are no nonzero ideals I, J of S such that
In 1963, Connell [Citation7, Theorem 8] gave a characterization of prime group rings. Indeed, given a unital ring A and a group G, the corresponding group ring is prime if, and only if, A is prime and G has no nontrivial finite normal subgroup.
Given a group G, recall that a ring S is said to be G-graded if there is a collection of additive subgroups of S such that
, and
, for all
. If, in addition,
, for all
, then S is said to be strongly G-graded. The class of unital strongly G-graded rings includes for instance all group rings, all twisted group rings, and all G-crossed products.
In 1984, Passman [Citation22, Theorem 1.3] generalized Connell’s result by giving a characterization of prime unital strongly group graded rings. In recent years, various generalizations of strongly group graded rings have appeared in the literature. In [Citation16], so-called nearly epsilon-strongly group graded rings were introduced. That class of rings contains for instance all unital strongly group graded rings, all epsilon-strongly group graded rings, all Leavitt path algebras, and all unital partial crossed products (see [Citation16, Citation17]). In [Citation9], a characterization of prime nearly epsilon-strongly group graded rings was established, thereby generalizing Passman’s result to a non-unital and non-strong setting.
In this paper, we turn our focus to rings graded by groupoids. Let G be a groupoid. Recall that a ring S is said to be G-graded if there is a collection of additive subgroups of S such that
, and
whenever
are composable, and
otherwise. Groupoid rings, groupoid crossed products, and partial skew groupoid rings are examples of rings that, by construction, are naturally graded by groupoids (see e.g. [Citation1, Citation2, Citation4, Citation5, Citation20]). Partial skew groupoid rings play a key role in the theory of partial Galois extensions for partial groupoid actions (see e.g., [Citation3, Theorem 5.3]) and, in particular, in the Galois theory of weak Hopf algebra actions on algebras (see [Citation6]). Some crossed product algebras defined by separable extensions are not, in a natural way, graded by groups, but instead by groupoids (see e.g. [Citation11, Citation12]). Another concrete example of when it can be beneficial to make use of a groupoid grading instead of just a group grading is in the study of Leavitt path algebras. There, one may utilize the canonical grading by the so-called free path groupoid which is finer, and encodes more of the structure, than the coarser canonical
-grading (see [Citation8]).
Suppose that G is a groupoid and that S is a G-graded ring. Following [Citation10], we shall say that S is nearly epsilon-strongly G-graded if, for each is an s-unital ring and
. Our main result provides a characterization of prime nearly epsilon-strongly groupoid graded rings.
Theorem 1.1.
Let G be a groupoid, let S be a nearly epsilon-strongly G-graded ring, and let . The following statements are equivalent:
S is prime;
is G-prime, and for every
is prime;
is G-prime, and for some
is prime;
S is graded prime, and for every
is prime;
S is graded prime, and for some
is prime;
For every
e is a support-hub, and
is prime;
For some
e is a support-hub, and
is prime.
Here denotes the isotropy group of an element
For more details about the statements in the above theorem, see e.g. Definitions 3.17 and 3.21.
We point out that our main result reduces the primeness investigation for a groupoid graded ring to the group case. Indeed, is a nearly epsilon-strongly group graded ring (cf. [Citation16]). Hence, the main result of [Citation9] can be used to decide whether it is prime.
Here is an outline of this paper. In Section 2, based on [Citation15, Citation19, Citation25], we recall some basic definitions and properties about groupoids, groupoid graded rings, and s-unital rings that will be used throughout the paper. In Section 3, we record some basic properties of nearly epsilon-strongly groupoid graded rings. Inspired by [Citation9], for such a ring S, we establish a relationship between the G-invariant ideals of and the G-graded ideals of S (see Theorem 3.16). Moreover, we provide necessary conditions for graded primeness of S (see Section 3.3) and establish our main result which is a characterization of prime nearly epsilon-strongly groupoid graded rings (see Theorem 1.1). Finally, in Section 4, we apply our results to partial skew groupoid rings, skew groupoid rings, and groupoid rings. In particular, we give a characterization of prime partial skew groupoid rings associated with groupoid partial actions of group-type [Citation4, Citation5] (see Theorem 4.15). Using that every global action of a connected groupoid is of group-type, we get a characterization of prime skew groupoid rings of connected groupoids (see Theorem 4.19). Furthermore, we establish a generalization of Connell’s classical result, by providing a characterization of prime groupoid rings (see Theorem 4.26).
2 Preliminaries
In this section, we recall some notions and basic notation regarding groupoids and graded rings.
2.1 Groupoids
By a groupoid, we shall mean a small category G in which every morphism is invertible. Each object of G will be identified with its corresponding identity morphism, allowing us to view G0, the set of objects of G, as a subset of the set of morphisms of G. The set of morphisms of G will simply be denoted by G. This means that .
The range and source maps , indicate the range (codomain) respectively source (domain) of each morphism of G. By abuse of notation, the set of composable pairs of G is denoted by
. For each
we denote the corresponding isotropy group by
Definition 2.1.
Let G be a groupoid.
G is said to be connected, if for every pair
, there exists
such that s(g) = e and r(g) = f.
A nonempty subset H of G is said to be a subgroupoid of G, if
and
whenever
and
.
2.2 Groupoid graded rings
Definition 2.2.
Let G be a groupoid. A ring S is said to be G-graded (or graded by G) if there is a collection of additive subgroups of S such that
, and
, if
, and
, otherwise.
Remark 2.3.
Suppose that G is a groupoid and that S is a G-graded ring.
If H is a subgroupoid of G, then
is an H-graded subring of S. In particular, note that
and
are subrings of S for every
.
For any element
, with
, we define
.
An ideal I of S is said to be a graded ideal (or G-graded ideal) if
.
The next lemma generalizes [Citation19, Lemma 2.4]. For the convenience of the reader, we include a proof.
Lemma 2.4.
Let G be a groupoid and let S be a G-graded ring. Suppose that H is a subgroupoid of G. Define by
The following assertions hold:
The map
is additive.
If
and
, then
and
.
Proof.
(i) This is clear.
(ii) Take and
. Put
. Clearly,
and
. If
and
, then either the composition gh does not exist or it belongs to
. Thus,
. Hence,
Analogously, one may show that
. □
2.3 s-unital rings
We briefly recall the definitions of s-unital modules and rings as well as some key properties.
Definition 2.5
([Citation15, cf. Definition 4]). Let R be a ring and let M be a left (resp. right) R-module. We say that M is s-unital if (resp.
) for every
. If M is an R-bimodule, then we say that M is s-unital if it is s-unital both as a left R-module and as a right R-module. The ring R is said to be left s-unital (resp. right s-unital) if it is left (resp. right) s-unital as a left (resp. right) module over itself. The ring R is said to be s-unital if it is s-unital as a bimodule over itself.
The following results are due to Tominaga [Citation25]. For the proofs, we refer the reader to [Citation15, Propositions 2.8 and 2.10].
Proposition 2.6.
Let R be a ring and let M be a left (resp. right) R-module. Then M is left (resp. right) s-unital if, and only if, for all and all
there is some
such that
(resp.
) for every
.
Proposition 2.7.
Let R be a ring and let M be an R-bimodule. Then M is s-unital if, and only if, for all and all
there is some
such that
for every
.
Remark 2.8.
The element a, in Proposition 2.7, is commonly referred to as an s-unit for the set .
3 Groupoid graded rings
Throughout this section, unless stated otherwise, G denotes an arbitrary groupoid.
3.1 Nearly epsilon-strongly groupoid graded rings
In this section, we will recall the notion of a nearly epsilon-strongly groupoid graded ring and record some of its basic properties.
Definition 3.1
([Citation10, Definition 3]). Let S be a G-graded ring. We say that S is nearly epsilon-strongly G-graded if, for each is an s-unital ring and
.
Remark 3.2.
The above definition simultaneously generalizes [Citation16, Definition 3.3] and [Citation18, Definition 34].
The following characterization of a nearly epsilon-strongly groupoid graded ring appeared in [Citation10, Proposition 15] without a proof. For the convenience of the reader, we provide it here.
Proposition 3.3.
Let S be a G-graded ring. The following statements are equivalent:
S is nearly epsilon-strongly G-graded;
For all
and
there exist
and
such that
Proof.
Suppose that (i) holds. Let and
We may write
for some
,
and
. Notice that
and
for every
By assumption,
and
are s-unital and hence, by Proposition 2.6, there exist
and
such that
and
for every
Thus,
This shows that (ii) holds.
Conversely, suppose that (ii) holds. Let Note that, by assumption, Sg is s-unital as a left
-module and
is s-unital as a right
-module. Let
We may write
for some
and
. By s-unitality of the left
-module Sg, and Proposition 2.6, there is some
such that
for every
Similarly, there is some
such that
for every
Hence,
and
. This shows that
is s-unital. Note that
Using that Sg is s-unital as a left
-module we get that
. Thus,
. This shows that (i) holds. □
Corollary 3.4.
Let S be a nearly epsilon-strongly G-graded ring and let a be a nonzero element of S. If , then there are elements
such that
and
.
The following result generalizes [Citation9, Proposition 2.13] from the group setting.
Proposition 3.5.
Let S be a nearly epsilon-strongly G-graded ring. The following assertions hold:
Se is an s-unital ring, for every
.
, for every
S is s-unital and
is an s-unital subring of S.
Suppose that H is a subgroupoid of G. Then
is a nearly epsilon-strongly H-graded ring.
is an s-unital ring, for every
The set
is a subgroupoid of G.
Proof.
Take
. By assumption,
Therefore,
and hence
.
Let
, with
Take
. By Proposition 3.3, there exist
and
such that
The set
is finite, because
is finite. For every
, by (i), Sf is s-unital, and we let
be an s-unit for the finite set
Define
We get that
Similarly, we define the finite set
For every
by (i), we let
be an s-unit for the finite set
Define
We get that
It follows immediately from (ii).
It follows immediately from Remark 2.3 and the fact that
.
Take
. Clearly, the isotropy group
is a subgroupoid of G. By (iv) and (iii) we get that
is s-unital.
Clearly,
whenever
. Suppose that
and
. Then
and
Therefore,
This shows that
is a subgroupoid of G.
Take
such that
. We claim that
. If we assume that the claim holds, then clearly
Now we show the claim. Let
be nonzero. By Proposition 3.3, there are
and
such that
In particular,
and
□
Remark 3.6.
(a) Note that (vi) above holds for any G-graded ring. For (vii), however, the nearly epsilon-strongness of the G-grading is used.
(b) Suppose that S is a nearly epsilon-strongly G-graded ring. By Proposition 3.5(vii), whenever
. The converse, however, need not hold (see e.g., Example 4.22).
Example 3.7.
Let be a groupoid with
and depicted as follows:
Let be the ring of 3 × 3 matrices over
and let
denote the standard matrix units. We define:
Notice that It is not difficult to see that the ring
is nearly epsilon-strongly G-graded.
3.2 Invariance in groupoid graded rings
Inspired by [Citation9, Sections 3–4], we shall now examine the relationship between G-graded ideals of a G-graded ring S and G-invariant ideals of the subring Throughout this section, S denotes an arbitrary G-graded ring.
Definition 3.8
([Citation9, Definitions 3.1 and 3.3]). Let S be a G-graded ring.
For any
and any subset I of S, we write
Let H be a subgroupoid of G and let I be a subset of S. Then, I is called H-invariant if
for every
Remark 3.9.
Note that if and
then
Lemma 3.10.
If and J is an ideal of
then Jg is an ideal of
Proof.
Let and let J be an ideal of
. Notice that Jg is an additive subgroup of
Moreover,
Analogously,
□
Proposition 3.11.
Suppose that J is an ideal of Then SJS is a G-graded ideal of S.
Proof.
It is clear that SJS is an ideal of S and that Now, we show the reversed inclusion. Take
and
If
then
Otherwise,
, and then
Thus,
□
Lemma 3.12.
Suppose that is s-unital and that J is an ideal of
Then J is G-invariant if, and only if,
Proof.
We first show the “only if” statement. Suppose that J is G-invariant. For each we have
Let . By Proposition 3.11, SJS is G-graded and we notice that
. Thus,
. By assumption,
is s-unital and
. Hence,
Now we show the “if” statement. Suppose that Take
and notice that
Thus, J is G-invariant. □
Lemma 3.13.
If I is a G-graded ideal of S, then is a G-invariant ideal of
Proof.
Let I be a G-graded ideal of S. Clearly, is an ideal of
Take
Notice that
Furthermore, if
then
Therefore,
□
Lemma 3.14.
Let S be a nearly epsilon-strongly G-graded ring. If I is a G-graded ideal of S, then
Proof.
Let I be a G-graded ideal of S. By Proposition 3.5, S is s-unital and hence Thus,
Analogously,
We claim that Take
and
By Proposition 3.3, there is some
such that
Then,
for some
and
Notice that
for every
Hence,
Using that I is G-graded, we get that
Similarly,
Thus,
and
□
Corollary 3.15.
Let S be a nearly epsilon-strongly G-graded ring. If J is a G-invariant ideal of then
Proof.
Let J be a G-invariant ideal of By Proposition 3.11, SJS is a G-graded ideal of S, and, by Lemma 3.14,
Thus, by Proposition 3.5(iii) and Lemma 3.12,
□
By Lemmas 3.13 and 3.11, the following maps are well defined:
The following theorem generalizes [Citation9, Theorem 4.7] and [Citation2, Theorem 3.12].
Theorem 3.16.
Let S be a nearly epsilon-strongly G-graded ring. The map defines a bijection between the set of G-graded ideals of S and the set of G-invariant ideals of
The inverse of
is given by
Proof.
Let I be a G-graded ideal of S. Lemma 3.14 implies that Let J be a G-invariant ideal of S. Notice that, by Proposition 3.5(iii) and Lemma 3.12,
□
3.3 Graded primeness of groupoid graded rings
In this section, we identify necessary and sufficient conditions for graded primeness of a groupoid graded ring.
Definition 3.17.
Let S be a G-graded ring.
is said to be G-prime if there are no nonzero G-invariant ideals I, J of
such that
.
S is said to be graded prime if there are no nonzero G-graded ideals I, J of S such that
.
The following result generalizes [Citation2, Proposition 3.29].
Theorem 3.18.
Let S be a nearly epsilon-strongly G-graded ring. Then S is graded prime if, and only if, is G-prime.
Proof.
We first show the “if” statement. Suppose that is G-prime and let I1, I2 be nonzero G-graded ideals of S. By Lemma 3.13 and Corollary 3.4,
and
are nonzero G-invariant ideals of
Then
Now, we show the “only if” statement. Suppose that S is graded prime and let J1, J2 be nonzero G-invariant ideals of By Proposition 3.5(iii) and Proposition 3.11,
and
are nonzero G-graded ideals of S. By Corollary 3.15 and our assumption,
Thus,
. □
Now, we determine some necessary conditions for graded primeness of a groupoid graded ring.
Lemma 3.19.
Let S be a G-graded ring. Then SbtS is a G-graded ideal of S for all and
Proof.
Take and
. Clearly,
is an ideal of S. Notice that
Now, take
, and
If
or
then
Otherwise, we have that
This shows that □
Lemma 3.20.
Let S be a G-graded ring which is s-unital. Suppose that S is graded prime. Let and
be nonzero elements, for some
. Then there is some
and
such that
is nonzero.
Proof.
We prove the contrapositive statement. Suppose that for all
and
Consider the sets
and
which, by Lemma 3.19 and the s-unitality of S, are both nonzero G-graded ideals of S. By assumption, we have
This shows that S is not graded prime. □
Definition 3.21.
Let S be a G-graded ring. An element (see Proposition 3.5(vi)) is said to be a support-hub if for every nonzero
, with
, there are
such that
r(k) = e, and
and
are both nonzero.
Remark 3.22.
Let S be a G-graded ring.
Suppose that
is a support-hub and that
is nonzero, for some
. Notice that there are
as in the following diagram.
Notice that, if S is a ring which is nearly epsilon-strongly graded by a group G, then the identity element e of G is always a support-hub.
Proposition 3.23.
Let S be a G-graded ring which is s-unital. If S is graded prime, then every is a support-hub.
Proof.
We prove the contrapositive statement. Suppose that there is some which is not a support-hub. Then there are
and a nonzero element
, such that for every
such that
we have that
or for every
such that
we have that
Let ae be a nonzero element of
Using Lemma 3.19 and the fact that S is s-unital,
and
are nonzero G-graded ideals of S.
Notice that if for every such that
we have that
then
Moreover, if for every
such that
we have that
then
Therefore, S is not graded prime. □
Proposition 3.24.
Let S be a G-graded ring which is s-unital. The following assertions hold:
If G is a connected groupoid, then
is a connected subgroupoid of G.
If there is a support-hub in
, then
is a connected subgroupoid of G.
If S is graded prime, then
is a connected subgroupoid of G.
Proof.
Suppose that G is connected. Take
By assumption, there is
such that s(g) = e and
Since Se and Sf are nonzero, we must have
, and hence
is connected.
Suppose that
is a support-hub. Take
By the definition of
there are nonzero elements
and
Since e is a support-hub, there is some
such that r(k) = e and
In particular,
Using again that e is a support-hub, there is some
such that s(h) = e and
Hence,
Define
and note that
and
It follows from Proposition 3.23 and (ii). □
3.4 Primeness of groupoid graded rings
In this section, we will provide necessary and sufficient conditions for primeness of a nearly epsilon-strongly G-graded ring. Furthermore, we will extend [Citation9, Theorem 1.3] to the context of groupoid graded rings.
Proposition 3.25.
Let S be a nearly epsilon-strongly G-graded ring. If S is prime, then is prime for every
Proof.
We prove the contrapositive statement. Let Suppose that I and J are nonzero ideals of
such that
. By Proposition 3.5(iii), S is s-unital and hence
and
are nonzero ideals of S. Clearly,
We claim that
If we assume that the claim holds, then it follows that
, and we are done. Now we show the claim. Take
and
. Let
If
or
, then
Otherwise,
and then, since I and J are ideals of
, we get that
Thus,
. □
Remark 3.26.
Let S be a nearly epsilon-strongly G-graded ring.
By Propositions 3.25 and 3.24(iii), if S is prime, then
is prime for every
and
is connected. The converse, however, need not hold as shown by Example 4.22.
Recall that, by Lemma 2.4,
is defined by
for every
.
The next result partially generalizes [Citation9, Lemma 2.19].
Lemma 3.27.
Let S be a nearly epsilon-strongly G-graded ring and let I be a nonzero ideal of S. If is a support-hub, then
is a nonzero ideal of
Proof.
Suppose that is a support-hub. By Lemma 2.4,
is an ideal of
We claim that
Let
be an element where all the homogeneous coefficients are nonzero and the
are distinct. By Corollary 3.4, there is some nonzero
such that
is nonzero and contained in
.
Notice that is nonzero and contained in I. Thus, without loss of generality, we may assume that
. Since e is a support-hub, there is an element
such that r(k) = e and
is nonzero. In particular, there is an element
such that
is nonzero. Using again that e is a support-hub, there is an element
such that s(h) = e and
is nonzero. Therefore, there is an element
such that
is nonzero. Hence,
and
Notice that if, and only if,
Thus,
□
Theorem 3.28.
Let S be a nearly epsilon-strongly G-graded ring. If there is some such that e is a support-hub and
is prime, then S is prime.
Proof.
Suppose that is a support-hub and that
is prime. Let I and J be nonzero ideals of S. By Lemma 3.27,
and
are nonzero ideals of
and hence, by assumption,
We claim that Let
and consider the finite set
If
, then
. Now, suppose that
and take
. Using that
there are
,
and
such that
By Proposition 2.6, using that Se is s-unital, there is some such that
for all
and all
Thus,
for all
and
Using a similar argument, there is some such that
for every
such that s(t) = e. Therefore,
Analogously, Thus,
and S is prime. □
Remark 3.29.
The assumption on the existence of a support-hub in Theorem 3.28 cannot be dropped. Indeed, consider the groupoid and the groupoid ring
. Then
and
. Furthermore,
and
are both prime. Nevertheless, S is not prime.
The next example shows that the existence of a support-hub in a connected grading groupoid is not enough to guarantee (graded) primeness of the graded ring.
Example 3.30.
Let be a groupoid with
and depicted as follows:
Define S as the ring of matrices over of the form
Denote by the standard matrix units and define:
and
otherwise. It is not difficult to verify that this G-grading is nearly epsilon-strong. Notice that
and that
is a support-hub. However, observe that
and
are nonzero elements and that there is no element
such that
and
Therefore, by Lemma 3.20, S is not graded prime.
Now, we prove our main result.
Proof of Theorem 1.1.
It follows from Proposition 3.25 and by the definition of primeness that (i) (iv)
(v). By Proposition 3.5(iii), S is s-unital and Proposition 3.23 implies (iv)
(vi)
(vii) and (v)
(vii). By Theorem 3.28, (vii)
(i). Finally, note that by Theorem 3.18, (ii) is equivalent to (iv), and (iii) is equivalent to (v). □
Remark 3.31.
In [Citation14], Munn investigates primeness of rings graded by inverse semigroups. He shows (see [Citation14, Theorem 4.1]) that if S is a so-called 0-bisimple inverse semigroup, R is a faithful restricted S-graded ring, and RG is prime for some nonzero maximal subgroup G of S, then R is prime. We point out that Munn’s theorem can potentially be used to prove e.g. (vi)
(i) in Theorem 1.1. Indeed, we may associate a natural inverse semigroup
with the groupoid G and view any G-graded ring as an S(G)-graded ring (see e.g., [Citation10, Section 4.3]). It is easy to come up with examples of prime nearly-epsilon strongly G-graded rings such that the corresponding S(G)-gradings fail to satisfy the requirements in Munn’s theorem. However, given a prime nearly epsilon-strongly G-graded ring R, it is not clear to the authors whether one can always find a subgroupoid H of G, contained in
, such that S(H) and its grading on R do in fact satisfy the requirements in Munn’s theorem.
We recall that Passman [Citation22] provided a characterization of prime unital strongly group graded rings. That result was generalized in [Citation9, Theorem 1.3] to nearly epsilon-strongly group graded rings.
Theorem 3.32
([Citation9, Theorem 1.3]). Let G be a group and let S be a nearly epsilon-strongly G-graded ring. The following statements are equivalent:
S is not prime;
There exist:
subgroups
,
an H-invariant ideal I of Se such that
for all
and
nonzero ideals
of SN such that
and
There exist:
subgroups
with N finite,
an H-invariant ideal I of Se such that
for all
and
nonzero ideals
of SN such that
and
There exist:
subgroups
with N finite,
an H-invariant ideal I of Se such that
for all
and
nonzero H-invariant ideals
of SN such that
and
There exist:
subgroups
with N finite,
an H-invariant ideal I of Se such that
for all
and
nonzero H/N-invariant ideals
of SN such that
and
Remark 3.33.
Note that, in Theorem 1.1, is nearly epsilon-strongly graded by the group
. Hence, one can use Theorem 3.32 to decide whether
is prime.
The following Theorem generalizes [Citation9, Theorem 1.4].
Theorem 3.34.
Let S be a nearly epsilon-strongly G-graded ring. Suppose that there is some such that
is torsion-free. Then S is prime if, and only if, Se is
-prime and
is G-prime.
Proof.
It follows from Theorem 1.1 and [Citation9, Theorem 1.4]. □
4 Applications to partial skew groupoid rings
In this section, we will apply our main results on primeness for nearly epsilon-strongly groupoid graded rings to partial skew groupoid rings, (global) skew groupoid rings, and groupoid rings. In particular, we will characterize prime partial skew groupoid rings induced by partial actions of group-type (cf. [Citation4]). Furthermore, we will generalize [Citation9, Theorem 12.4] and [Citation9, Theorem 13.7].
Throughout this section, unless stated otherwise, G denotes an arbitrary groupoid and A denotes an arbitrary ring.
4.1 Partial skew groupoid rings
Definition 4.1.
A partial action of a groupoid G on a ring A is a family of pairs satisfying:
For each
is an ideal of A, Ag is an ideal of
, and
is a ring isomorphism,
, for every
,
, whenever
,
, for all
and
.
Definition 4.2.
Given a partial action σ of a groupoid G on a ring A one may define the partial skew groupoid ring as the set of all formal sums of the form
, where
is zero for all but finitely many
, and with addition defined point-wise and multiplication given by
Remark 4.3.
(a) Throughout this section, unless stated otherwise, will assume that is an arbitrary partial action of G on A, that Ag is an s-unital ring, for every
, and that
. As a consequence,
will always be an associative ring (see [Citation2, Remark 2.7 (ii)]), and there will exist a ring isomorphism (cf. [Citation2, Lemma 3.6 (i)])
defined by
(1)
(1)
(b) Under the above assumptions, by [Citation2, Lemma 3.2], Ag is an ideal of A, for every
(c) It is readily verified that any partial skew groupoid ring carries a natural G-grading defined by letting
, for every
.
The following result generalizes [Citation9, Proposition 13.1].
Proposition 4.4.
The partial skew groupoid ring is a nearly epsilon-strongly G-graded ring.
Proof.
Let . Using that
is s-unital, and hence idempotent, we get that
Now, using that Ag is s-unital we get that is s-unital, and that Ag is idempotent. Hence,
This shows that is nearly epsilon-strongly G-graded. □
Remark 4.5.
By Propositions 3.5 and 4.4, the partial skew groupoid ring is s-unital.
Definition 4.6.
Let G be a groupoid, let A be a ring and let be a partial action of G on A.
Let H be a subgroupoid of G. An ideal I of A is said to be H-invariant if
for every
.
A is said to be G-prime if there are no nonzero G-invariant ideals I, J of A such that
.
The next result generalizes [Citation9, Remark 13.4] from the group setting.
Proposition 4.7.
Let I be an ideal of A. Then I is G-invariant in the sense of Definition 4.6 if, and only if, is a G-invariant ideal of
in the sense of Definition 3.8. In particular, A is G-prime if, and only if,
is G-prime.
Proof.
Suppose that I is an ideal of A. Let By Remark 4.3(b) and the s-unitality of Ag, we get that
and
. Furthermore,
Notice that
. We get that
Therefore,
□
Remark 4.8.
Recall that, with the natural G-grading on
, an element
is a support-hub if for every nonzero element
, with
, there are
such that
r(k) = e and both
and
are nonzero.
For
, denote by
the partial action of the isotropy group
on the ring Ae, obtained by restricting σ. The associated partial skew group ring is denoted by
.
Theorem 4.9.
Let be a partial action of G on A such that Ag is s-unital for every
and
. Then, the following statements are equivalent:
The partial skew groupoid ring
is prime;
A is G-prime and, for every
is prime;
A is G-prime and, for some
is prime;
is graded prime and, for every
is prime;
is graded prime and, for some
is prime;
For every
e is a support-hub and
is prime;
For some
e is a support-hub and
is prime.
Proof.
It follows from Proposition 4.4, Theorem 1.1, and Proposition 4.7. □
We recall the following result from [Citation9]. In that paper, the authors say that a partial skew group ring is s-unital if it is defined by a partial group action on s-unital ideals.
Theorem 4.10
([Citation9, Theorem 13.7]). Let G be a group and let be an s-unital partial skew group ring. Then,
is not prime if, and only if, there are:
subgroups
with N finite,
an ideal I of A such that
for every
for every
and
(iii) nonzero ideals
of
such that
and
for every
Remark 4.11.
Note that, in Theorem 4.9, is an s-unital partial skew group ring. Thus, one can apply Theorem 4.10 to determine whether
is prime.
Definition 4.12
([Citation4, Remark 3.4]). A partial action of a connected groupoid G on a ring A is said to be of group-type if there exist an element
and a family of morphisms
in G such that
and
for every
Remark 4.13.
If a partial action σ is of group-type (and hence G is connected), then every element of G0 can take the role of e in the above definition (see [Citation4, Remark 3.4]).
By [Citation4, Lemma 3.1], every global action by a connected groupoid is of group-type. The converse does not hold. For an example of a non-global partial action of group-type, we refer the reader to [Citation4, Example 3.5].
Lemma 4.14.
Let be a partial action of G on A such that Ag is s-unital for every
and
. Furthermore, let
and consider the following statements:
σ is of group-type (and G is connected);
For every nonzero element
there is some
such that
r(k) = e and
For every nonzero element
there is some
such that
r(k) = e and
e is a support-hub.
Then, (i) (ii)
(iii)
(iv).
Proof.
(i) (ii) Suppose that (i) holds. Let
and
By Remark 4.13(a), since σ is of group-type, there is a morphism
such that
and
Note that
Define
and the proof is done.
(ii) (iii) Suppose that (ii) holds. Let
and
By assumption, there is some
such that
r(k) = e and
Since
is s-unital, we get
(iii) (iv) Suppose that (iii) holds. Let
and
By assumption, there is some
such that
and r(k) = e and
Hence, there is some
such that
Let
be an s-unit for
and let
be an s-unit for
Note that
and,
Define and note that
. Moreover, we have that
and
are both nonzero. This shows that e is a support-hub.
(iv) (iii) Suppose that (iv) holds. Let
and
By assumption, there is some
such that r(k) = e and
is nonzero. Note that
Therefore,
□
Theorem 4.15.
Let be a partial action of a connected groupoid G on A such that Ag is s-unital for every
and σ is of group-type. Then the partial skew groupoid ring
is prime if, and only if, there is some
such that
is prime.
Proof.
It follows from Lemma 4.14 and Theorem 4.9. □
Now, we will make use of the example from [Citation4, Example 3.5] of a non-global partial action of a connected groupoid on a ring, of group-type, and apply Theorem 4.15 to it.
Example 4.16.
Let be the groupoid with
and the following composition rules:
We present in the following diagram the structure of G:
Let be the field of complex numbers and let
where
and
We define the partial action
of G on A as follows:
and
where
denotes the complex conjugate of a, for all
By choosing
and
we notice that σ is of group-type (cf. Definition 4.12).
Now, we describe the group partial action of
on
Note that
We claim that Ae is not -prime. Let
and
Note that I and J are nonzero
-invariant ideals of Ae and
By Theorem 4.9,
is not prime. An analogous argument shows that
is not prime. Hence, Theorem 4.15 implies that
is not prime.
The following result generalizes [Citation9, Theorem 13.5] from the group setting.
Theorem 4.17.
Let be a partial action of G on A such that Ag is s-unital for every
and
Furthermore, suppose that there is some
such that
is torsion-free. Then
is prime if, and only if, Ae is
-prime and A is G-prime.
Proof.
It follows from Propositions 4.4, 4.7, and Theorem 3.34. □
4.2 Skew groupoid rings
The partial action of G on A is said to be global if
for every
. In that case, the corresponding partial skew groupoid ring
is said to be a skew groupoid ring (see e.g., [Citation20, Citation21]).
Remark 4.18.
Let be a global action of G on A.
For
is a ring isomorphism.
Note that
, whenever
.
The multiplication rule on the skew groupoid ring is induced by the following somewhat simplified rule compared to the partial case:
Theorem 4.19.
Let be a global action of G on A such that Ae is s-unital for every
, and let
. Suppose that the groupoid G is connected. Then, the skew groupoid ring
is prime if, and only if, there is some
such that
is prime.
Proof.
It follows from Remark 4.13(b) and Theorem 4.15. □
Proposition 4.20.
Let be a global action of G on A such that Ae is s-unital for every
, and let
. The following statements are equivalent:
is connected;
For every
e is a support-hub;
For some
e is a support-hub.
Proof.
Obviously, (ii) (iii).
(iii) (i) This follows from Proposition 3.24(ii).
(i) (ii) Suppose that
is connected. Take
, and let
be a nonzero element. Then
and, by assumption, there is some
such that
and
Notice that
By Lemma 4.14 (ii)
(iv), e is a support-hub. □
Below, we summarize our findings for skew groupoid rings.
Theorem 4.21.
Let be a global action of G on A such that Ae is s-unital for every
, and let
. Then, the following statements are equivalent:
The skew groupoid ring
is prime;
A is G-prime, and for every
is prime;
A is G-prime, and for some
is prime;
is graded prime, and for every
is prime;
is graded prime, and for some
is prime;
is connected, and for every
is prime;
is connected, and for some
is prime.
Proof.
It follows from Theorem 4.9 and Proposition 4.20. □
The following example shows that Theorem 4.21 does not generalize to partial skew groupoid rings.
Example 4.22.
Let be a groupoid such that
s(g) = f and r(g) = e as follows:
Let Now, we define a partial action
of G on A:
and
and
Notice that and
are prime rings. Observe that
, and that G is connected. However, there are no
and
such that
. Hence, by Lemma 3.20,
is not graded prime.
Corollary 4.23.
Let be a global action of G on A such that Ae is s-unital for every
, and let
. Suppose that there is some
such that
is torsion-free. Then, the skew groupoid ring
is prime if, and only if, Ae is
-prime and
is connected.
Proof.
It follows from Theorem 4.21 and [Citation9, Theorem 13.5]. □
4.3 Groupoid rings
Let R be an s-unital ring and let G be a groupoid. The groupoid ring consists of elements of the form
where
is zero for all but finitely many
. For
and
, the multiplication in
is defined by the relation
, if g, h are composable, and
otherwise.
Remark 4.24.
Let R be an s-unital ring and let G be a groupoid. Consider the global action of G on A, defined by letting
and
for every
, and
. Notice that the corresponding skew groupoid ring
is isomorphic to the groupoid ring
.
A subset is said to be R-dense if for every nonzero
there is some
such that
For each
define
.
Proposition 4.25.
Let G be a groupoid and let R be a nonzero s-unital ring. The following statements are equivalent:
For every
is R-dense;
There is some
such that
is R-dense;
G is connected.
Proof.
(i) (ii) The proof is immediate.
(ii) (iii) Suppose that there is some
such that
is R-dense. Let
and let
be nonzero. Clearly,
and hence, by assumption,
By the definition of
, we may find some
such that s(g) = f and
(iii) (i) Fix
and suppose that G is connected. Clearly,
which is R-dense. □
The following theorem generalizes [Citation9, Theorem 12.4].
Theorem 4.26.
Let G be a groupoid and let R be a nonzero s-unital ring. The following statements are equivalent:
The groupoid ring
is prime;
G is connected and there is some
such that the group ring
is prime;
G is connected and, for every
the group ring
is prime;
There is some
such that
is R-dense and
is prime;
For every
is R-dense and
is prime;
G is connected, R is prime and there is some
such that
has no non-trivial finite normal subgroup;
G is connected, R is prime and, for every
has no non-trivial finite normal subgroup;
R is prime, and there is some
such that
is R-dense, and
has no non-trivial finite normal subgroup;
R is prime, and for every
is R-dense, and
has no non-trivial finite normal subgroup.
Proof.
Notice that The proof follows from Remark 4.24, Theorem 4.21(i), (vi), and (vii), Proposition 4.25 and [Citation9, Theorem 12.4]. □
Remark 4.27.
(a) It is known that, in the case where R is a commutative unital ring, the groupoid ring is an example of a Steinberg algebra (see [Citation23, Remark 4.10]). Hence, in that special case, the equivalence between (i) and (iv) in Theorem 4.26 can be obtained using Steinberg’s results from [Citation24, Proposition 4.3], [Citation24, Proposition 4.4], and [Citation24, Theorem 4.9].
(b) In the case where R is unital, after suitable translations of the properties involved, it is possible to obtain e.g. the implication (ii)
(i) in Theorem 4.26 from [Citation13, Theorem 3.2].
Acknowledgments
The authors are grateful to Patrik Lundström for making them aware of Munn’s result in [Citation14].
Disclosure statement
The authors report that there are no competing interests to declare.
Additional information
Funding
References
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