Vol. 3(Spec 1) (2014) , 343–355 © Palestine Polytechnic University-PPU 2014
ON THE NOTION OF ’RETRACTABLE MODULES’ IN THE
CONTEXT OF ALGEBRAS
Christian Lomp
Dedicated to Patrick Smith and John Clark on the occasion of their 70th birthdays. Communicated by Jawad Abuhlail
MSC 2010 Classifications: Primary 16N60, 16D10; Secondary 16S40, 16W30.
Keywords and phrases: retractable modules, algebras with large centre, fix rings, Hopf algebras, endomorphism rings. This research was funded by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT - Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2011.
Abstract. This is a survey on the usage of the module theoretic notion of a “retractable module" in the study of algebras with actions. We explain how classical results can be interpreted using module theory and end the paper with some open questions.
1 Introduction
This note is written for module theorists and intends to show where the module theoretic notion of a retractable module plays a role in the context of algebras with certain additional structure. These "additional structures" include group actions, involutions, Lie algebra actions or more gen-erally Hopf algebra actions as well as the bimodule structure of the algebra (and combinations of all these). Such algebraAis usually a subalgebra of a larger algebraBand has the structure of a cyclic leftB-module, while its endomorphism ring EndB(A) is isomorphic to a subalgebra AB ofA. In this intrinsic situation the condition onAto be a retractableB-module means that
the subringAB has non-zero intersection with all non-zero left ideals ofAthat are stable under
the module action ofB. We will first recall the module theoretical notion of a retractable module and set it in a categorical and lattice theoretical context. In the second section we will examine various situations of algebrasAwith additional structures and recall manytheorem classical theo-rems that can be expressed in terms of module theory. The last section deals with open problems around retractable modules in the context of algebras. Note that all rings/algebras are considered to be associative and unital. Modules are usually meant to be left modules and homomorphisms are acting from the right.
2 Module Theory
A retractable module is a (left)A-moduleM, over some ringA, such that there exist non-zero ho-momorphisms fromM into each of its non-zero submodules. The notion of a retractable module appeared first in the work of Khuri [17] and had since then been used in connection with primness conditions and the nonsingularity of a module and its endomorphism ring (see [12, 14–16, 33]). One of Khuri’s result is the establishment of a bijective correspondence between closed submod-ules of a moduleM and closed left ideals of its endomorphism ringS = EndA(M) in caseM
is non-degenerated (see [13, 33, 34]). A module is non-degenerated if its standard Morita con-text is non-degenerated. Recall that a Morita concon-text between two ringsAandS is a quadruple (A, M, N, S) whereAMS andSNAare bimodules with bimodule maps (−, −) :M ⊗SN → A
and [−, −] : N ⊗AM → S satisfying m[n, m0] = (m, n)m0 and n(m, n0) = [n, m]n0 for all m, m0 ∈ Mnon-degenerated and n, n0 ∈ N. The context is called non-degenerated ifMS is
faithful and for all 06=m ∈ Malso [N, m]6= 0. The standard Morita context of a leftA-module
M is the context (A, M, M∗, S) withS= EndA(M) andM∗= HomA(M, A) and the maps
(−, −) :M ⊗SM∗→ A (m, f) := (m)f ∀m ∈ M, f ∈ M∗ (2.1)
MS is obviously always faithful and M is non-degenerated if and only if [M∗, m] 6= 0 for
all 0 6= m ∈ M. In this case there exist for any non-zero submoduleN ofM and non-zero elementm ∈ N a homomorphism f : M → Asuch that the map ˜f : M → N with (n) ˜f = (n)f m ∈ Am ⊆ N, forn ∈ M, is non-zero. Hence any non-degenerated module is in particular retractable.
Retractable modules have gained recently further attention in [8–10, 18, 19, 26–28], but have previously also played a major role in the context of algebras. The theorems of Bergman-Isaacs or Rowen say that in certain situations an algebraAwith a group actionGor considered as bi-module is a retractable bi-module considered over the skew group ringA ∗ Gor over the enveloping algebraAe=A ⊗ Aop. In case∂is a derivation on an algebraA, thenAis a retractableA[x;∂
]-module if∂is locally nilpotent. Furthermore Cohen’s question raises the problem as to whether a semiprime algebraAwith an action of a semisimple Hopf algebraH is a retractable A#H -module. With this in mind, the survey was written to illustrate the use of the module theoretic notion of retractability in the context of algebras.
2.1 Categorical notions
A retractable module is clearly a generalisation of a self-generator. Here we shortly review this notion in the context of category theory.
Definition 2.1. LetC be a category. An objectX ofCis generated by an object GofC if for every pair of distinct morphismsf, g : X → Y inCthere exists a morphismh: G → X with
hf 6=hg.
In particular ifCis an Abelian category andXis not the zero object, then Mor(G, X)6={0}, because for the identityf =idXand the zero morphismg= 0, there exist a morphismh:G → Xsuch thath 6= 0. Having this in mind the definition of a retractable object seems to be a direct generalisation of a generator.
Definition 2.2. An objectM of an Abelian categoryCis called retractable if Mor(M, N)6={0}
for all subobjectsNofM, different from the zero object.
LerC be an Abelian category with arbitrary coproducts. LetM be any object inCandN a subobject of it. Then there exists a unique subobject Tr(M, N) ofN such that every morphism
f : M → N factors through Tr(M, N). Suppose thatM is a retractable object, then Tr(M, N) is essential inN for each non-zero subobjectN ∈ Lin the sense that for all non-zero subobjects
KofNthe meetK ∩Tr(M, N) is non-zero (as anyf :M → Kcan be considered a morphism
f : M → N and hence factored through Tr(M, N)). This means in the case of a module categoryC, that a moduleM is retractable if and only if for all submodulesN ofM, the trace Tr(M, N), which is the sums of images of all homomorphismsf : M → N, is essential in
N. Loosely speaking every submodule of a retractable moduleM can be "approximated" by an
M-generated submodule.
2.2 Lattice theoretical meaning
LetRbe ring andM a leftR-module with endomorphism ringS = End(M). To link module theoretical properties ofM with properties ofS one can use the following map from the lattice
L(M) of leftR-submodules ofM to the latticeL(S) of left ideals ofS:
Hom(M, −) :L(M)→ L(S) N 7→Hom(M, N) ={f ∈ S |(M)f ⊆ N }.
This map Hom(M, −) is always a homomorphism of semilattices between (L(M), ∩) and (L(S), ∩) since Hom(M, N ∩ L) = Hom(M, N)∩Hom(M, L) holds for allN, L ∈ L(M). Call a homomorphismϕ:L → L0 of semilattice with least element 0 faithful ifϕ(x) = 0⇒ x= 0.
The following Lemma can be proven easily: Lemma 2.3. LetM be a leftA-module.
(i) Hom(M, −) is injective if and only ifM is a self-generator. (ii) Hom(M, −) is faithful if and only ifM is a retractable module.
While the injectivity or faithfulness of Hom(M, −) has to do withM being a generator or retractable, the surjectivity of Hom(M, −) deals with the projectivity ofM (we refer the reader to [30] for all undefined notion.):
Lemma 2.4. LetM be a leftA-module andS= EndA(M).
(i) All cyclic left ideals ofS belong to the range of Hom(M, −) if and only if M is semi-projective.
(ii) All finitely generated left ideals ofSbelong to the range ofHom(M, −) if and only ifM is intrinsically-projective.
(iii) Hom(M, −) is surjective ifM is Σ-self-projective, i.e. projective in the Wisbauer category
σ[M].
Considering semi-projective retractable modules combines the light generator and self-projectivity condition onM. Such modules were studied for example in [9].
3 Algebras with additional structures
An (associative, unital) algebraAover a commutative ringRis anR-moduleAsuch that there existR-linear maps
µ:A⊗RA → A and η:R → A,
called the multiplication ofAand unit ofArespectively, such thatAwithµas multiplication, defined asab=µ(a ⊗ b) fora, b ∈ A, and 1A=η(1) as unit element forms an associative and
unital ring.
It is easy to see that by taking R = Z, any (associative, unital) ring is an (associative,
unital) algebra over Z. Thus for those that do not like the idea of R-algebras, they might for the beginning just ignore R and think ofA being and ordinary ring. Clearly η does not need to be injective, just think of A = Zn[x], for some n > 1, which is a Z-algebra and η : Z → Zn ⊆ Zn[x] is the canonical map, where Zn = Z/nZ. Moreover the image ofη lies always in the centre of A and in particular Ais an R0-algebra for any subring R0 of the centreZ(A) ={a ∈ A | ab=ba ∀b ∈ A}.In the following letRbe always a commutative ring andAanR-algebra.
3.1 Algebras that are retractable as bimodule
The endomorphism ring EndR(A) of Aas R-module is itself an R-algebra whose R-module
structure is given as follows: for allr ∈ R, f ∈EndR(A) setr · f:A → Aby (r · f)(x) :=rf(x)
for allx ∈ A. The multiplication of EndR(A) is given by the composition of functions and the
unit map is given byη:R →EndR(A) sendingr 7→ r ·idA.
For each elementa ∈ Athere are twoR-linear maps ofAwhich are the left and the right multiplication bya:
la:A → A la(x) :=ax ∀x ∈ A, ra:A → A ra(x) :=xa ∀x ∈ A.
Note that since the multiplication ofA is supposed to be associative,la and rb commute, i.e. la◦ rb=rb◦ lain EndR(A), for anya, b ∈ A. The subalgebra of EndR(A) generated by all maps la andrbfora, b ∈ A. Is called the multiplication algebra ofAand denoted byM(A).
LeftM(A)-modulesM can be considered as bimodules over A, where one defines for all
a, b ∈ Aandm ∈ M:
am:=la• m and mb:=rb• m.
The bimodule compatibility condition (am)b = a(mb) for allm ∈ M holds, because of (rb◦ la − la ◦ rb)• M = 0. Analogously any A-bimodule has a natural structure as left M(A
)-module given byla• m = amand rb• m = mb, fora, b ∈ A andm ∈ M. The enveloping
algebraAe = A⊗
RAop is also an R-algebra, whereAop denotes the opposite ring ofA. The
multiplication ofAeis defined as
(a ⊗ x)(b ⊗ y) := (ab)⊗(yx) ∀a, b, x, y ∈ A.
Moreover the mapψ:Ae→End
R(A) given by
a ⊗ b 7→ la◦ rb ∀a, b ∈ A
is a surjective algebra map fromAetoM(A) whose kernel is the annihilator ofA, whereAis naturally considered a leftAe-module by (a ⊗ b)• x=axbfor alla, b, x ∈ A. Hence
Ker(ψ) = ( n X i=1 ai⊗ bi ∈ Ae| n X i=1 aixbi= 0 ∀x ∈ A ) = AnnAe(A).
For a bimoduleM ∈ Ae-Mod one defines its centre as
Z(M) ={m ∈ M | am=ma ∀a ∈ A}.
The canonical map
ψM : HomAe(A, M)−→ Z(M) given by f 7→(1)f ∀f ∈HomAe(A, M). (3.1)
is a bijection, where the leftAe-module homomorphism is applied from the right. The inverse of this map is
ψM−1:Z(M)−→HomAe(A, M) given by m 7→[a 7→ a · m] ∀m ∈ Z(M). (3.2)
In particularψA : EndAe(A) ' Z(A) is an isomorphism ofR-algebras and the bijectionsψM
are leftZ(A)-linear maps.
Lemma 3.1.Ais a retractable leftAe-module if and only ifAhas a large centreZ(A), i.e. every
non-zero two-sided ideal ofAcontains a non-zero central element.
Proof. This follows from the the fact that theAe-submodules ofAare precisely the two-sided
idealsIand from the bijection
ψI : HomAe(A, I)' Z(I) =I ∩ Z(A).
There are at least two important results that have to be mentioned in this context. The first is due to Rowen and says the following (see [25]):
Theorem 3.2 (Rowen, 1972). Any semiprime PI algebra has a large centre.
Recall that anR-algebraAis a PI-algebra if it there exists an elementf(x1, . . . , xn) in the free
algebraRhx1, . . . , xnioverRsuch thatf(a1, . . . , an) = 0 for any substitutiona1, . . . , an ∈ A
and such that one of the coefficients of a monomial of highest degree of f is 1. Examples of semiprime PI-algebras are matrix algebras over division algebras that are finite dimensional over their centre or more generally any semiprime algebra that is a finitely generated when considered a module over its centre. Thus Rowen’s theorem says that any semiprime PI-algebra is a retractable leftAe-module. For a non-trivial example one might consider the quantum plane
at root of unity. Letq ∈C\ {0}. The quantum plane over C with parameterqis the algebra
A= Cq[x, y] = Chx, yi/hyx − qxyi.
Elements ofAcan be uniquely written as linear combinations of monomials of the formxiyj
fori, j ≥ 0. The relationyx = qxy makes Cq[x, y] a non-commutative algebra ifq 6= 1. An
elementary calculation shows that the centre of Cq[x, y] isZ(A) = C ifqis not a root of unity
and that it is Z(A) = C[xn, yn] if q is a primitive n-th root of unity. In the later caseA is
generated by all monomials of the formxiyjwith 0≤ i, j < nas a module overZ(A). Hence Ais a PI-algebra. SinceAis also a domain the centre is large, i.e. any non-zero ideal of Cq[x, y]
contains a non-zero polynomial of the formf(xn, yn).
The second result in this context is Puczyłowski and Smoktunowicz’ description of the Brown-McCoy radical of an algebraAfrom [23]. Recall that the Brown-McCoy radical BMc(A) ofAis the intersection of all maximal two-sided ideals. This means that the Brown-McCoy rad-ical is the module theoretic radrad-ical ofAas bimodule, i.e. BMc(A) = Rad(AeA). Puczyłowski
and Smoktunowicz described the Brown-McCoy radical ofA[x] using the following ideal:
P S(A) =\{P ⊆ A | P is a prime ideal andA/P has a large centre} .
Theorem 3.3 (Puczyłowski-Smoktunowicz, 1998). BMc(A[x]) =P S(A)[x]. Their result relies on the following (surprising) Lemma from [23]:
Lemma 3.4. LetM be a maximal ideal ofA[x]. IfA ∩ M = 0, thenAhas a large centre. In the case of the Lemma, if such maximal idealM ofA[x] exist withM ∩ A= 0, thenA
will also be a prime ring. Recall that a moduleM over some ringAis called compressible if
M embeds into any non-zero submodule of it, i.e. for any 06=N ⊆ M there exists an injective
A-linear mapf :M → N. Prime algebras with large centre are precisely the algebras that are compressible as bimodule.
Lemma 3.5 (see [30, 35.10]). An algebraAis a compressibleAe-module if and only if it is prime and has a large centre.
Hence, in module theoretic terms, P S(A) is the intersection of all thoseAe-submodules P
3.2 Derivations
A (R-linear) derivation of anR-algebraAis an R-linear map∂ : A → Asuch that ∂(ab) =
∂(a)b+a∂(b) for alla, b ∈ A. Examples are ordinary partial derivations∂xi on the polynomial
ringR[x1, . . . , xn] overR. For anya ∈ Aof anR-algebraA, its commutator∂(a) = [a, −], with
[a, b] =ab − baforb ∈ A, is a derivation, called an inner derivation ofA.
Given a derivation ∂ one constructs the differential operator ring B = A[x;∂] as the R -algebra generated byAandxsubject to
xa=ax+∂(a) ∀a ∈ A.
The algebraA[x;∂] can be constructed as a subalgebra of EndA(A[x]) such thatA[x;∂] is a free
leftA-module with basis{xi| i ∈
N}. Hence the elements ofBcan be uniquely written as (left) polynomialsPn
i=0aixiwithai∈ A. MoreoverAbecomes a leftA[x;∂]-module with respect to
the actionx · a=∂(a) or more generally
n X i=0 aixi ! · b= n X i=0 aiδi(b) ∀b ∈ A, ∀ n X i=0 aixi∈ B.
The leftA[x;∂]-submodules ofAare precisely those left idealsIofAthat are stable under the derivation, i.e.∂(I)⊆ I. For any leftA[x;∂]-moduleM one defines its submodule of constants as
M∂ ={m ∈ M | x · m= 0}= AnnM(x).
ForM =Aone hasA∂ = Ker(∂) which is easily seen to be a subalgebra ofA. Analogously to the bimodule situation we have the followingR-linear isomorphisms for any leftA[x;∂]-module
M:
ψM : HomA[x;∂](A, M)−→ M∂ given by f 7→(1)f ∀f ∈HomA[x;∂](A, M).
(3.3) Its inverse map is
ψM−1:M∂ −→HomA[x;∂](A, M) given by m 7→[a 7→ a · m] ∀m ∈ M∂. (3.4)
In particularψA : EndA[x;∂](A) ' A∂ is an isomorphism ofR-algebras and the bijectionsψM
are leftA∂-linear maps. Using these isomorphisms the following Lemma is obvious:
Lemma 3.6.Ais a retractableA[x;∂]-module if and only ifA∂ is large inA, i.e. A∂ intersects
all non-trivial∂-stable left ideals ofAnon-trivially.
A sufficient condition for this to happen is the local nilpotency of∂, i.e. if for everya ∈ A, there existsn ∈N such that∂n(a) = 0.
Proposition 3.7. If∂is locally nilpotent, thenAis a retractableA[x;∂]-module.
Proof. The proof of this fact is obvious, because if 0 6= a ∈ I is a non-zero element of an∂ -stable left idealIofA, then by hypothesis there existsn ∈N such that∂n(a) = 0. Take the least n ∈N such that∂n(a) = 0, then∂n−1(a) is a non-zero element ofI ∩ A∂, which proves thatA∂
is large inAand henceAis a retractableA[x;∂]-module. For example the partial derivatives ∂
∂xi ofA = R[x1, . . . , xn] for any i = 1, . . . , nare
lo-cally nilpotent. However it is unknown whenAis a retractableA[x;∂]-module for an arbitrary derivation∂.
Question 3.8. What are necessary and sufficient conditions for Ato be a retractable A[x;∂ ]-module? in other words, find conditions onAand∂such that any non-zero ∂-stable left ideal contains a non-zero constant.
Zelmanowitz called a leftR-module fully retractable if for any non-zero submoduleN and non-zerog:N → M there existsh:M → Nsuch thathg 6= 0. It is not clear whenAis fully retractable asA[x;∂]-module. The next Proposition can be found in [6].
Proposition 3.9 (Borges-Lomp, 2011). Let∂be a locally nilpotent derivation ofA. (i) Ais a compressible leftA[x;∂]-module, providedA∂ is a domain.
(iii) Ais critically compressible as leftA[x;∂]-module if and only ifA∂ is a left Ore domain andAis fully retractable as leftA[x;∂]-module.
LetR=kbe a field of characteristic zero and∂a locally nilpotent derivation onAsuch that there exists an elementa ∈ Awith∂(a) = 1. Then by [6, Proposition 3.10]Ais a self-projective
A[x;∂]-module and module theory yields the following result (see [6, Proposition 3.12]) Proposition 3.10 (Borges-Lomp, 2011). LetAbe an algebra over a fieldkof characteristic zero and∂a locally nilpotent derivation ofAsuch that∂(a) = 1, for somea ∈ A. Then the following statements are equivalent:
(a) A∂ is a left Ore domain; (b) Ais a left Ore domain;
(c) Ais a critically compressibleA[x;∂]-module.
Example 3.11 (Goodearl, 1980). LetA =k[[t]] be the power series ring over a fieldkof char-acteristic 0 and let∂be the derivation with∂(tn) =ntnfor alln ≥0. Certainly∂is not locally
nilpotent. Any ideal of Ais ∂-stable, because the proper ideals are of the formI = Atn for n ≥0. Since for anya=P∞
n=0antn ∈ Aone has∂(a) =P∞n=0nantn 6= 1 we have that there
does not exist anya ∈ Awith∂(a) = 1. NeverthelessAis a self-projective leftA[x;∂]-module. To see this note that by the correspondence (3.3) it is enough to show that (A/I)∂ = (A∂+I)/I
for any∂-stable left idealI ofA. Let I = Atm be any ideal ofAand a = P∞
n=0antn with ∂(a)∈ I. Then there existsb ∈ Asuch that∂(a) =P∞
n=0nantn =btm∈ I, which implies that ai = 0 for all 1≤ i < m. Thusa = a0+b0tmfor someb0 ∈ Aanda+I = a0+IinA/I.
Sincea0∈ A∂ this shows (A/I)∂ ⊆(A∂+I)/I while the reversed inclusion is obvious. Since Ais a Noetherian integral domain,A[x;∂] is a left Noetherian Ore domain. However asA∂ =k
is the base field andAis not simple as leftA[x;∂]-module,Ais not retractable and hence not compressible asA[x;∂]-module.
Note that the set DerR(A) of derivations on theR-algebraAforms a Lie algebra with the
ordinary Lie product induced by the product(=composition) of EndR(A), i.e. if∂, ∂ ∈DerR(A),
then their commutator
[∂, δ] =∂ ◦ δ − δ ◦ ∂ ∈DerR(A)
is again a derivation ofA. An action of an arbitrary abstract Lie algebra g overRby derivations is given by a map of Lie algebrasd: g→DerR(A). We shall write the image ofx ∈g underd
asdx. An analogous construction to the construction of the differential operator ring is given by
a new product on the tensor product ofAand the universal enveloping algebraU(g) of g. The new algebra is called the smash product ofAandU(g) and is denoted byA#U(g). The product is determined by
(1#x)(a#1) =a#x+dx(a)#1 ∀x ∈g, a ∈ A.
Later we will shortly mention Hopf algebras and their action on rings andU(g) is one of the examples. AgainAbecomes a leftA#U(g)-module where the module action is given by (a#x)· b=a dx(b) for alla, b ∈ Aandx ∈ g. Again one can consider the subset of all those elements a ∈ Asuch thatdx(a) = 0 for allx ∈g, i.e.
Ag= \
x∈g
Ker(dx).
For an arbitrary leftA#U(g)-moduleM one setsMg=T
x∈gAnnM(1#x).As before there are
functorialRlinear isomorphismsMg'HomA#U (g)(A, M) and in particularAg'EndA#U (g)(A).
Retractability forAas a leftA#U(g)-module also means here thatAgis large inAwith respect to all those left ideals ofAthat are stable under alll derivationsdxwithx ∈g.
In the case of a single derivation∂ ∈ DerR(A) one considers the trivial Lie algebra g = R
with zero bracket. The mapd : R → DerR(A) is then given byr 7→ r∂ for allr ∈ R. Note
that the enveloping algebra of the trivial Lie algebra is the polynomial ringR[x] in one variable. Moreover
A#U(g) =A⊗RR[x] =A[x]
is determined by the product:
(1#x)(a#1) =a#x+∂(a)#1 or better xa=ax+∂(a) ∀a ∈ A,
3.3 Group Actions
A groupGact on anR-algebraAby automorphism, which means that there is a homomorphism of groupsρ:G →AutR(A) fromGto the group ofR-linear automorphisms ofA. We denote the
image ofg ∈ Gunderρbyρg, although we sometimes writegainstead ofρg(a) fora ∈ Aand g ∈ G. As in the last section, a new algebra can be attached toGandA, which is the skew-group ringA ∗ Gdefined onA⊗RR[G], whereR[G] is the group ring ofGoverR. Alternatively one
might considerA∗Gas the free leftA-module with basis{g | g ∈ G}such that the multiplication is defined as
ag · bh=aρg(b)gh=a(gb)gh ∀a, b ∈ A, ∀g, h ∈ G.
IfGis cyclic infinite, i.e. G = hσi, thenA ∗ Gis equal to the Laurent skew-polynomial ring
A[x, x−1;σ] whose underlying space are the Laurent polynomials with coefficients in A and whose multiplication is determined by
xna=σn(a)xn ∀a ∈ A, n ∈Z.
IfG=hσiis cyclic of ordern, thenA ∗ Gis equal to the factorA[x;σ]/hxn−1iof the skew-polynomial ringA[x;σ].
LetGbe any group acting onAas automorphism. ThenAhas a leftA ∗ G-module structure defined by
ag · b=aρg(b) ∀a, b ∈ A, g ∈ G.
The leftA ∗ G-submodules ofAare precisely theG-stable left ideals ofA. LetM be any left
A ∗ G-module. Then the submodule of fixed elements ofM is
MG ={m ∈ M | ∀g ∈ G: g · m=m.}
Moreover one has againR-linear isomorphisms for any leftA ∗ G-moduleM:
ψM : HomA∗G(A, M)−→ MG given by f 7→(1)f ∀f ∈HomA∗G(A, M).
(3.5) with inverse map
ψ−1M :MG−→HomA∗G(A, M) given by m 7→[a 7→ a · m] ∀m ∈ MG. (3.6)
In particularψA: EndA∗G(A)' AGis an isomorphism ofR-algebras and the bijectionsψM are
leftAG-linear maps.
Lemma 3.12.Ais a retractableA ∗ G-module if and only ifAGis large inA, i.e. AGintersects all non-trivialG-stable left ideals ofAnon-trivially.
The existence of non-trivial fixed elements inG-stable left ideals reduces the study of the structure ofG-stable left ideals ofAto the study of left ideals ofAG. The following result is a
classical theorem in the study of group action:
Theorem 3.13 (Bergman-Isaacs, 1973; Kharchenko, 1974). LetGbe a finite group of ordern
acting on anR-algebraA. Assume that one of the following conditions is verified: (i) Aisn-torsionfree and does not contain any non-zero nilpotentG-stable ideal or (ii) Ais reduced, i.e. does not contain any nilpotent element.
ThenAis retractable as leftA ∗ G-module.
For the proof of (i) see [3, 22] for the proof of (ii) see [11].
3.4 Involutions
LetAbe anR-algebra. An involution ofAis anR-linear map ∗ : A → Awitha 7→ a∗ that is an anti-algebra homomorphism and has order 2, i.e. (ab)∗ = b∗a∗ and (a∗)∗ = a for all a, b ∈ A. An elementa ∈ Ais called symmetric (respectively anti-symmetric) with respect to
∗ if a∗ = a(respectively. a∗ = −a). Ideals that are stable under the involution∗ are called ∗-ideals. Consider the subalgebraB of EndR(A) generated by∗and all left multiplications la
fora ∈ A, i.e.
Note that for anya, b ∈ Aone has
ra(b) =ba= (a∗b∗)∗= (∗ ◦ la∗◦ ∗)(b).
Hencera = ∗ ◦ la∗◦ ∗ ∈ B. It is clear that Abecomes a left B-module by simply applying
f ∈ B ⊆ EndR(A), i.e. for anyf ∈ Banda ∈ Asetf · a := f(a). The leftB-submodules
ofAare stable under left and right multiplicationsla andra for anya ∈ Aand hence are
two-sided ideals ofA. Moreover they are stable under∗. On the other hand any∗-ideal is also a left
B-submodule.
The algebraBcan be seen as a factor of a skew group algebra. Let Ae = A⊗RAop be the
enveloping algebra ofAand consider the mapα:Ae → Aedefined byα(a ⊗ b) =b∗⊗ a∗for
alla, b ∈ A. The mapαis an automorphism ofAe, because for anya, b, c, d ∈ A:
α((a ⊗ b)(c ⊗ d)) =α(ac ⊗ db) = (db)∗⊗(ac)∗=b∗d∗⊗c∗a∗= (b∗⊗ a∗) (d∗⊗ c∗) =α(a⊗b)α(c⊗d).
Asαis its own inverse it is an automorphism of order 2. LetG=hαi ={id, α}and consider the (surjective) mapψ:Ae∗ G → Bgiven by (a ⊗ b)⊗ id+ (c ⊗ d)⊗ α 7→ l
a◦ rb+lc◦ rd◦ ∗
for alla, b, c, d ∈ A. Thenψis an algebra homomorphism. The calculations are easy but tedious and will be illustrated on the example of the product of (1⊗1)⊗ αand (a ⊗ b)⊗ id. Note first that for anyx ∈ A:
lb∗◦ ra∗◦ ∗(x) =b∗x∗a∗= (axb)∗=∗ ◦ la◦ rb(x).
Hence
ψ((1⊗1⊗ α)(a ⊗ b ⊗ id)) =ψ(b∗⊗ a∗⊗ α) =lb∗◦ra∗◦∗=∗◦la◦rb=ψ(1⊗1⊗ α)(a ⊗ b ⊗ id)).
ThusBis a factor algebra ofAe∗ G. For any leftAe∗ G-moduleM one defines the submodule
of central symmetric elements as
Z(M;∗) =Z(M)∩ MG
={m ∈ Z(M)| α · m=m}.
ForM =Aone obtains the central symmetric elements ofA, i.e.Z(A;∗) ={a ∈ Z(A)| a∗= a}. Again one hasR-isomorphisms for any leftAe∗ G-moduleM:
ψM : HomAe∗G(A, M)−→ Z(M;∗) given by f 7→(1)f ∀f ∈HomAe∗G(A, M).
(3.7) with inverse map
ψM−1:Z(M;∗)−→HomAe∗G(A, M) given by m 7→[a 7→ a · m] ∀m ∈ Z(M;∗).
(3.8) In particularψA : EndAe∗G(A) ' Z(A;∗) is an isomorphism ofR-algebras and the bijections
ψM are leftZ(A;∗)-linear maps.
Lemma 3.14.Ais a retractableAe∗ G-module if and only if every non-zero∗-ideal contains a
non-zero central symmetric element.
Using Rowen’s theorem we have the following:
Corollary 3.15. Let∗be an involution of anR-algebraA. IfAis a semiprime PI-algebra, then
Ais a retractableAe∗ G-module.
Proof. LetI be a non-zero∗-ideal. By Rowen’s Theorem 3.2, I contains a non-zero central element, saya ∈ I. SinceIis∗-stable,a∗∈ I. Thusa+a∗is a central symmetric element ofI. Ifa+a∗= 0, thena∗=−aanda2is a central symmetric element as (a2)∗= (−a)2=a2.Note
thata26= 0 asais non-zero and central andAis semiprime.
4 Open Problems
If a groupGacts on an algebraAby automorphisms, thenAbecomes a module over the skew group ring A ∗ G as well as over the group algebra R[G]. If a Lie algebra g acts on A by derivations, thenA becomes a module overA#U(g) as well as over the universal enveloping algebraU(g). Both algebrasR[G] andU(g) are examples of Hopf algebras and the constructions
that there exist algebra maps ∆ :H → H ⊗ H(called the comultiplication ofH) and:H → R
(called the count ofH) such that the following diagrams commute:
H ∆ // ∆ H ⊗ H ∆⊗1 H ⊗ H 1⊗∆ // H ⊗ H ⊗ H R ⊗ H oo ⊗1 H ⊗ H 1⊗ // H ⊗ R H ' ee ' 99 ∆ OO
For an elementh ∈ H its comultiplication ∆(h) is an element of H ⊗ H. It is common to use the so-called Sweedler’s notation enumerating symbolically the first and second tensorand by writing ∆(h) =P
(h)h1⊗ h2∈ H ⊗ H.
The ring of R-linear endomorphisms of EndR(H) of a Hopf algebra H has another ring
structure as the usual given by the convolution product which associates to two endomorphisms
f, gofHthe endomorphismsf ∗ g=µ ◦(f ⊗ g)◦∆ whereµdenotes the multiplication ofH. To obtain a Hopf algebra one also requires that the identity map has an inverse in EndR(H) with
respect to the convolution product. This inverse is usually denoted bySand called the antipode ofH. Equivalently there should exist an endomorphismSsatisfying
µ ◦(id ⊗ S)◦∆ =η ◦ =µ ◦(S ⊗ id)◦∆
whereη : R → H denotes the mapr 7→ r1H for allr ∈ R. A Hopf algebraH acts on an R-algebraAifAis a leftH-module and an algebra in the category of leftH-modules. The later means that the multiplicationm : A⊗RA → A and the unit mapR → Awithr 7→ r1A are
maps of leftH-modules. Note that due to the comultiplication the category of leftH-modules is closed under tensor products, i.e. it is a tensor category. For left H-modulesN and M, elements x ∈ N and y ∈ M and h ∈ H with ∆(h) = P
(h)h1 ⊗ h2 ∈ H ⊗ H one sets h ·(x ⊗ y) =P
(h)(h1· x)⊗(h2· y).The base ringRbecomes a leftH-module byh · r=(h)r
for allh ∈ H, r ∈ R. Hence forH to act onA,Ahas to be a leftH-module and the following conditions have to be fulfilled for alla, b ∈ Aandh ∈ H.
h ·(ab) =X
(h)
(h1· a)(h2· b) and h ·1A=(h)1A.
The smash product ofAandH is denoted byA#H and defined on the tensor productA⊗RH
with multiplication given by
(a ⊗ h)·(b ⊗ g) =X
(h)
a(h1· b)⊗ h2g ∀a, b ∈ A, h, g ∈ H.
ThenAbecomes a cyclic leftA#H-module by the action (a ⊗ h)• b=a(h · b) for alla, b ∈ A,
h ∈ H. For any leftA#H-moduleM one defines the submodule ofH-invariants ofM as
MH={m ∈ M | h · m=(h)m ∀h ∈ H}.
As in the previous sections one has a for any leftA#H-moduleM canonical maps:
ψM : HomA#H(A, M)−→ MH given by f 7→(1)f ∀f ∈HomA#H(A, M).
(4.1) with inverse map
ψM−1:MH−→HomA#H(A, M) given by m 7→[a 7→ a · m] ∀m ∈ MH. (4.2)
In particularψA: EndA#H(A)' AHis an isomorphism ofR-algebras and the bijectionsψMare
leftAH-linear maps.
For a groupGand its group algebraH = R[G], the Hopf algebra structure ofH is given by the comultiplication ∆(g) = g ⊗ g, the counit (g) = 1 and the antipode S(g) = g−1, for allg ∈ G. The group algebraH acts onA if there is a module actionH ⊗ A → A, say by
h ⊗ a 7→ h · a, for allh ∈ H, a ∈ Aand the two conditions from above are satisfied, i.e.
g ·(ab) = (g · a)(g · b) and g ·1A=(g)1A= 1A ∀g ∈ G.
Hence if one denotes byαgthe mapa 7→ αg(a) =g · a, then one sees from this two conditions
thatαgis an endomorphism of rings. SinceGis a group andAis supposed to be a leftH-module, αg−1=α−1g for allg ∈ G. It is easy to see thatA ∗ Gequals the smash productA#R[G].
For a Lie algebra g and its universal enveloping algebraH =U(g) , the Hopf algebra struc-ture ofH is given by the comultiplication ∆(x) = 1⊗ x+x ⊗1, the counit(x) = 0 and the antipodeS(x) = 0, for all x ∈ g. The Hopf algebraH acts onA if there is a module action
H ⊗ A → A, say byh ⊗ a 7→ h · a, for allh ∈ H, a ∈ Aand the two conditions from above are satisfied, i.e.
x ·(ab) = (1· a)(x · b) + (x · a)(1· b) =a(x · b) + (x · a)b and x ·1A= 0 ∀x ∈g.
Hence if one denotes by∂xthe mapa 7→ ∂x(a) =x · a, forx ∈g, then one sees from this two
conditions that∂xis a derivation ofA. Hence the Hopf algebraU(g) acts onAif each element
acts as a derivation onA. The converse also holds. In particular if∂is a single derivation onA, g=Ris the trivial Lie algebra andU(g) =R[x] is its universal enveloping algebra, one obtains an action ofH=R[x] onAby setting for any polynomialf(x) =Pn
i=0rixiand elementa ∈ A: f(x)· a=Pn
i=0ri∂i(a).Looking at the smash productA#R[x] one sees that the multiplication
coincides with that of the differential operator ringA[x;∂], because
(1A⊗ x)(b ⊗1H) = 1H(1H· b)⊗ x1H+ 1H(x · b)⊗1H1H=b ⊗ x+∂(b)⊗1H.
IdentifyingA⊗RR[x] withA[x] asR-modules, we obtain our usual multiplication rule.
4.1 Cohen’s problem
Miriam Cohen asked in 1985 whether the smash productA#H is a semiprime ring providedA
is semiprime andH is a semisimple Hopf algebra acting onA(see [7]). The questions is still open up to my knowledge. The semisimple condition onH implies thatAis a projectiveA#H -module. Hence the mapϕ : A#H → Awitha#h 7→ a(h) splits by the mapψ : A → A#H
such thate= (1A)ψis an idempotent in (A#H)H. Moreover for anyH-stable left idealIofA
one has that (I)ψ = (I#1)eis a left ideal ofA#H isomorphic toI. IfA#H is semiprime and
Aprojective asA#H-module then (I#1)e(I#1)ewould be non-zero for any non-zeroH-stable left idealI ofA. Hence 0 6= e(I#1)e = e(I)ψ = (e • I)ψshows thate • I is non-zero. As
e ∈(A#H)Hone also has 06=e • I ∈ AH∩ I 'Hom
A#H(A, I).
Corollary 4.1. If A#H is semiprime andA is projective as leftA#H-module, thenA is a re-tractable leftA#H-module.
Of course this is true much more generally, namely for torsionless modules over semiprime rings, due to Amitsur (see [1, Theorem 27, Corollary 21]):
Lemma 4.2 (Amitsur). Any leftA-module over a semiprime ringAthat can be embedded into a direct product of copies ofAis retractable.
Hence if Cohen’s question would have a positive answer, then A semiprime andH being semisimple acting onAwould imply thatAis a retractableA#H-module.
Question 4.3. LetH be a semisimple Hopf algebra acting on a semiprime algebraA. IsAa retractable leftA#H-module, or in other words, does every non-zeroH-stable left ideal intersect
AHnon-trivially ?
Recalling Bergman and Isaacs Theorem 3.13 we see that an answer to the question above would generalise their Theorem and would be "half way" towards a positive answer to Cohen’s question. For more on Cohen’s question I would like to refer to my recent survey [21].
4.2 Primness of endomorphism rings
There are many ways to carry the idea of a prime ring to modules. Bican et al. in [4] defined a product on the lattice of submodulesLof a leftA-moduleM, by defining for anyN, K ∈ L:
N ∗ K:=NHomA(M, K) = X
f:M →K
(N)f.
Together with this product,Lbecomes a partially ordered groupoid, i.e.Lis a partially ordered set by inclusion and the binary operation∗satisfiesN ∗K ⊆ L∗KandK ∗N ⊆ K ∗L, whenever
K, L, N ∈ Land N ⊆ L. Note that in general the∗-product is not associative. The notion of a prime element is naturally carried over to any partially ordered groupoid (see [5]), i.e. P ∈ L
is a prime element in the partially ordered groupoid (L, ∗) (see [4, 19]). By definition, ifM is
∗-prime, then 06=M ∗ N =MHomA(M, N) for all 06=N ⊆ M. HenceM is retractable. On
the other hand ifM is retractable and EndA(M) is a prime ring, then it is not difficult to see that M is∗-prime (see [19, 4.1]). However it had been left open whether in general a∗-prime module must have a prime endomorphism ring.
Question 4.4. Does a∗-prime module have a prime endomorphism ring ?
Several positive results were obtained in [2]. In particular if the endomorphism ring ofM is commutative, then the answer is yes (see [2, 2.3]). This applies in particular to the case where
M = Ais an algebra considered as a left Ae-module. As End
Ae(A) ' Z(A) is the centre of
A, we have thatAis∗-prime as leftAe-module if and only ifAhas a large centre which is an
integral domain. An analogous statement hold for algebras with involution. Furthermore it has been shown in [2] that if a moduleM is semi-projective or not singular, then the answer is also yes. Thus if a semisimple Hopf algebraH acts onA, thenAis projective as leftA#H-module andAis a∗-prime leftA#H-module if and only ifAH is large inAand also a prime ring. This
applies also to the case whereH =R[G] withGa finite group acting onAsuch thatn =|G|
is invertible inR. Another instance where the general module theoretic result can be applied is in case of a locally nilpotent derivation∂onAover a fieldRof characteristic 0 such that there existsx ∈ Awith∂(x) = 1. ThenAis a self-projectiveA[x;∂]-module and thusAis∗-prime as leftA[x;∂]-module if and only ifA∂ is a prime ring (since the locally nilpotency of∂implies the retractability ofA). However in general it is unknown whetherA∂ is always prime ifAis a ∗-prime leftA[x;∂]-module.
Question 4.5. Let∂ be any derivation onA. Is it true thatA∂ is a prime ring providedAis a ∗-primeA[x;∂]-module?
Zelmanowitz’ weakly compressible modulesM are precisely those with 0 being a semiprime element in the partially ordered groupoid (L, ∗), i.e. ifN ∗ N ⊆0, thenN= 0. This means that for any non-zero submoduleNofM there exists a homomorphismf :M → Nwith (N)f 6= 0. Clearly the notion of weakly compressible modules generalise the notion of∗-prime modules as well as the notion of compressible modules. Furthermore weakly compressible modules are ob-viously retractable. Alternatively a moduleM is called semiprime if any essential submodule of
McogeneratesM. Having in mind that∗-prime modulesM can be characterised by the property that any non-zero submoduleNofM cogeneratesM, one sees that also semiprime modules are a generalisation of∗-prime modules. It is not difficult to see that weakly compressible modules are semiprime (see [19, 5.5]) and it is an open question whether the converse holds:
Question 4.6. Is a semiprime module weakly compressible?
This question has been considered in [29] and remotely also in [27] and it seems that it is important to know whether semiprime modules are retractable. Hence we might ask:
Question 4.7. Are semiprime modules retractable?
What can be said about the cases mentioned above, e.g. ifM =AandH is a Hopf algebra acting onA?
4.3 Zelmanowitz’ problem
A critically compressible module is a compressible module that cannot be embedded into any of its proper factor modules. The following question arose in Zelmanowitz’ papers [31, 32]: Question 4.8. Is a compressible uniform module whose nonzero endomorphisms are monomor-phisms a critically compressible module?
Attempts to answer this question have been made in [24] where it has been shown that the hypothesis of the question are equivalent toM being a uniform retractable module whose endo-morphism ring is a domain. Furthermore the following result has been shown in [6, Proposition 3.1].
Proposition 4.9 (Borges-Lomp, 2011). Let M be a left A-module with endomorphism ringS
and self-injective hull Mc. ThenM is critically compressible if and only ifM is retractable, S = End(M) is a left Ore domain and End(Mc) = Frac(S) is the left division ring of fractions
Thus a negative answer to question 4.8 could be obtained through an example of a uniform retractable module whose endomorphism ring is a domain, but not a left Ore domain. We finish by asking this question for our algebra situations above. All mentioned examples were of the following form (see also [20]): LetAbe an algebra overR. LetBbe a subalgebra of EndR(A)
that contains all left multiplicationslafora ∈ A. ThenAbecomes a leftB-module by evaluating
endomorphisms, i.e.b ∈ B, a ∈ A:b•a:=b(a) and moreoverAcan be considered a subalgebra ofBbya 7→ lafor alla ∈ A. The mapα:B → Agiven byb 7→(b)α=b •1 for allb ∈ Bis a
surjective map of leftB-modules, i.e.Ais a cyclic leftB-module, and furthermore the inclusion
a 7→ la letsαsplit as left A-modules, because (la)α = la(1) = a for all a ∈ A. Any left B-moduleM can be naturally considered a leftA-module and there exists a map
ψM : HomB(A, M)−→ M given by f 7→(1)f ∀f ∈HomB(A, M). (4.3)
The image ofψM can be identified with the subset
Im(ψM) ={m ∈ M | b • m= (b)αm= (b •1)m}=:MB
which we call the submodule ofB-invariants ofM. In particularAB ' End
B(A) is naturally
isomorphic to the endomorphism ring ofAas leftB-module. As in the cases above one also has an inverse ofψM which is:
ψM−1:MB −→HomB(A, M) given by m 7→[a 7→ a • m] ∀m ∈ MB. (4.4)
As before one has thatAis a retractable leftB-module if and only ifAB is large inA, i.e. AB
intersects allB-stable left ideals ofA.
Question 4.10. Suppose thatAB is a domain and large inA. IfAis a left uniformB-module, does it follow thatAB is a left Ore domain.
In caseB=M(A) as in subsection 3.1 respectively in caseBis the subalgebra of EndR(A)
generated by the left multiplication and an involution ∗as in subsection 3.4, one obtains that
AB = Z(A) respectivelyAB = Z(A;∗) is commutative. Since a commutative domain is an Ore domain, the question above is obviously answered. However in case of a group action or an action by a derivation,AB might be non-commutative and raises the following questions:
Question 4.11. Let∂be a derivation onAsuch thatA∂ is a domain which is large inA. IsA∂
left Ore ifAis a uniform leftA[x;∂]-module?
Letσbe an automorphism ofA. ThenAG ={a ∈ A | σ(a) =a}=:Aσ, whereG=hσi.
Question 4.12. Letσbe an automorphism such thatAσis a domain which is a large inA. IsAσ
left Ore ifAis a uniform leftA[x;σ]-module (respectively leftA[x±1;σ]-module)?
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Author information
Christian Lomp, Department of Mathematics, FCUP, University of Porto, Rua Campo Alegre 687, 4169-007 Porto, Portugal.
E-mail:clomp@fc.up.pt
Received: February 14, 2014. Accepted: April 3, 2014.
