Math. Z. 158, 285-294 (1978) Nlathematisehe Zeitschrift by Springer-Verlag 1978
Primitive Ideals in Crossed Products and Rings with Finite Group Actions
Martin Lorenz Fachbereich Mathematik der Universit~t Essen, Universit~itsstr.3, D-4300 Essen, Federal Republic of Germany
Introduction
The set Priv(R) of all primitive ideals of the ring R(~I) can be topologized by declaring the subsets of the form {Pc Priv(R)l P = I}, I an ideal of R, to be closed ([9], Chap. 9). The topological space Priv(R) is a T:space iff all primitive ideals of R are maximal. This holds trivially in the case of commutative rings. It is also true (but not trivial) if R is a group algebra of a finitely generated nilpotent group or an enveloping algebra of a finite dimensional nilpotent Lie algebra (Zalesskij [14], Dixmier [61). For the moment, let ~ denote the class of rings whose primitive ideals are maximal. ~ is obviously stable under homomorphisms and is stable under Morita equivalence (cf. f.i. [11, p. 258/259). In general the property ~ is not inherited by subrings and overrings as easy examples show. In Section 1 of this note we consider the situation R cR~ [G], where R~ [G] denotes a crossed product of the finite group G over the ring R (1.1). Crossed products can be considered as a generalization of group rings. They are a useful tool for the description of certain factors of group algebras (cf. f.i. [14]) and have been studied in a number of papers, notably by Bovdi (f.i. [51). In Theorem (1.7) we prove that the property ~11 is inherited from R~ [G] to R and that the converse is true if [GI-I~R or ifG is finite solvable. As a by-product of the proof we obtain some results on the primitivity of R and R~ [G1. Section 2 deals with the following situation: R is a ring (associative with 1), G a finite group of automorphisms of R, and R e the fixed subring of R. We always assume that [G I- 1e R. Examining the behaviour of R-modules under restriction to R e we obtain the analogue of (1.7):
Reg-~l if and only if R~Yaa (Theorem 2.7).
A similar result for prime ideals has been shown by Fisher and Osterburg ([81, Theorems 4.2 and 4.5). The techniques developed to prove Theorem (2.7) can be used to deduce results on the primitivity of R and R e (2.8). In addition, they easily
0025-5874/78/0158/0285/$02.00 286 M. Lorenz yield Montgomery's theorem [-10] on the Jacobson radical of the fixed ring: J(R~)=J(R) ~ R G (2.5). The corresponding result also holds for the socle of R G if R is semiprime: soc (R ~) = soc (R) c~ R a (2.6). Finally, we show that R is a Jacobson ring (this means each prime ideal is an intersection of primitives), if R Gis a Jacobson ring. This result has also been announced by Armendariz [2]. All rings considered in this note contain a unit element. The unspecified word "module" means "unitary right module" and "ideal" stands for "two-sided ideal". Mod-R denotes the category of right R-modules. U(R) is the group of invertible elements of R, and Int (R) the group of all inner automorphisms of R.
1. Crossed Products
(1.1) For the reader's convenience and in order to fix our notations, we recall the definition of a crossed product of a group G over a ring R: Given maps c~: G ~ Aut(R) and 7: Gx G-+ U(R) such that (i) ~(x, y) ~(x y, z)= ~(y, z)~-1 ~(x, y z) and (ii) 7(x, y) r~(XY)-~=r~(Y)-i~(x)-~ o/(x, y). for all x,y, z~G, r6R, we define the crossed product R~ [G] to be the set of all formal sums of the form ~ rx2 with r~R, and t~=0 for almost all x~G. The x~G addition in R~ [G] is defined componentwise and the multiplication is given by the rule
(rx ~)(r, y)=r~ r; (~-~ 7(x, y) ~yy.
This makes R~EGj an associative ring with unit element ?(1, 1) -~ 1. (1.2) For later reference we gather some simple properties of the crossed product R~ [G], which follow easily from the definitions. (a) The map r~-*rT(1, 1)-11 is a ring monomorphism of R into R~ [G]. We therefore consider R as a subring of R~ [G]. (b) It follows from (ii) that the composite map G ~ Aut(R) -*Aut(R)/Int(R) is a homomorphism. Hence the set {x~G[ a(x) Mnt(R)} is a normal subgroup of G. (c) If N is a subgroup of G, then the restrictions ,' und ~2' of, and 7 to N (resp. N x N) define a crossed product R~; [N]. R~', IN] is a subring of R~ [G] in a canonical way. Furthermore R~ [G] is free as a left R~', [N]-module: a basis is given by {YiliEl} where {y~[ieI} is a full set of coset representatives for N in G. If N is normal in G, then R~ I-G] carries the structure of a crossed product of GIN over R~: IN]. (d) Let I be an ideal of R such that I ~(~ = I for all x E G and let/~: -- R/I. Then and 7 give rise to maps ~: G-~ Aut(/~)_ and y: G x G--+ U(/~) which obviously satisfy (i) and (ii). The crossed product R~ [-G] is isomorphic to R~ [G]/IR~ [G]. Crossed Products and Rings with Finite Group Actions 287
(1.3) The following lemma is an appropriate version of Clifford's classical restriction theorem. The usual proof (see f.i. [11], 7.2.16) works with the obvious notational changes.
Lemma. Let G be a finite group, R a ring and U:=R~[G] a crossed product. If Vs Mod- U is irreducible, then the restricted module VR contains an irreducible submodule W and we have
VR= ~ Wff. x~G
Let T:= {x~GI W,Y~ W} and let V1 be the sum of all submodules X of VR such that X ~ W. Then T is a subgroup of G and V1 is an irreducible U'-module (U':=R~I[T], (1.2) (c)) such that
V~VI@U. U'
(1.4) A ring S is called projective relative to the subring R (or R-projective) if the following holds: Each exact sequence O-->A-+B-*C~O of S-modules which splits when considered as a sequence in Mod- R splits in Mod-S.- The following lemma is based on the well-known Maschke averaging process.
Lemma. Let G be a finite group, R a ring such that IGl-~R and U:=R~[G] a crossed product. Then U is R-projective. Proof. Let A be a submodule of BeMod-U such that A R is a direct summand of B R. This means there is an R-homomorphism h':B-~A such that h'(a)=a for all a~A. Let
h(b)=lG1-1 ~ h'(bX-1)N. x~G
Then h is a U-homomorphism from B to A such that h(a)=a for all a~A. (1.5) Lemlna. Let R be a ring, G a finite group and T a subgroup of G such that [T[-I~R. Let U:=R~ [G] be a crossed product and U'.'=R~; [T], where ~' and 7' denote the restrictions of c~ and 7 (1.2)(c). If V is an irreducible U-module, then the restricted module Vv, is completely reducible of finite length. Pro@ (1) If X~Mod-R is irreducible, then X@ U' is completely reducible of R finite length: Let (,) O~A--+X@U'-*B-*O be exact in Mod-U'. The R sequence (,) splits as an R-sequence since X@ U'I~ is a finite direct sum of con- R jugates of X and hence completely reducible. Therefore, by Lemma (1.4), the sequence (,) splits in Mod - U'. X @ U' is clearly of finite length. R (2) VR @ U' is completely reducible of finite length: By (1.3), VR is completely R reducible of finite length. Therefore, by step (1), VR @ U' is completely reducible of finite length. R 288 M. Lorenz
(3) The assertion of the lemma now follows from the fact that Vv, is a homo- morphic image of VR @ U'. R (1.6) The next lemma is well-known (see [-3]). Lemma. Let R be a simple ring, G a finite group and let U: = R~ [G] be a crossed product such that the automorphisms e(x), 1 ~:x6G, are outer. Then U is simple. (1.7) Theorem. Let R be a ring, G a finite group and U: = R~ [-G] a crossed product. If all primitive ideals of U are maximal, then the primitive ideals of R are maximal. If IGI-I~R or G is finite solvable, then the converse holds. Proof. The proof of the first assertion is analogous to that of Snider [-13], Lemma 2: Suppose we are given ideals P, M of R such that P~M and P is primitive. Then since G is finite, we have P.'= ~ P~(X)~M:= 0 M~(X). xEG xaG Fix V~Mod-R irreducible such that P=annn(V ) and let W.'= V@ U. The R module W is obviously of finite length and hence we may fix a finite composition series W= Wo~ WI ~... ~ VV~=(O). Let Qi:=annv(Wi_l/Wi). Then Q1Q2 ... Qs cannv(W)=PU~MU. Since 1VTU is a proper two-sided (!) ideal of U, we can choose a maximal ideal N of U containing MU and obtain QicN for some i. Since Wn= ~ |174 it follows that Wz_I/W~ contains a copy V| for some x~G x~G and hence being U-invariant contains a copy of V| for each x~G. We conclude that
annR(Wi_l/Wi)=Qic~ R c ~ annR(V| ~ annR(V)~(~)=P ~M c N c~ R x~G x~G and hence Qz ~ N. Therefore Qz is a nonmaximal primitive ideal of U. The proof of the second assertion is by induction on ]G[. The case [G[=I being trivial, we assume that ]GI > 1 and that the assertion is true for groups of smaller order. In particular using (1.2)(c) we may assume that G is simple. Now let P~Q be primitive ideals of U. We have to show that P=Q. Let P=annv(V), Q= annv(W), where V, W~Mod-U are irreducible. The restricted module VR contains an irreducible submodule L (1.3). Set T..={x~G[L~Y~-L} and U':= R~', [,T] (~ U). We treat the two cases (i) T.< G and (ii) T= G separately. Case (i). By (1.3) we have V~-VI@ U where V1 is an irreducible U'-module. U' Let P1.'= annv,(V1). The restriction Wv, of W is completely reducible of finite length. In case ]GI I~R this follows from (1.5) and when G is finite solvable it follows from the fact that in this case T=(1) and U'=R. Let Wv,=WI|174 W~EMod- U' irreducible, and let Q~: = annv,(W//). Then
~ Qi=annv,(W)=Q c~ U'~ P c~ U'~ P1. i=l Crossed Products and Rings with Finite Group Actions 289
Therefore QicP1, say, and by induction ([TI annv(Wi @ U) = annv(V1 @ U) = annv(V ) = P. U' U' The embedding of W~ into W extends to a U-homomorphism W~@ U~ W U' which has to be epi since W is irreducible. Hence annv(W~@ U)~annv(W)=Q, U' that is P ~ Q and we are done in case (i). Case (ii). Now VR is isomorphic to a finite direct sum of copies of L (1.3) and hence Pc~R=anng(L ) is a maximal ideal of R. Furthermore by (1.3) and by assumption on R we obtain Q c~ R = (~ M ~(~ for some maximal ideal M of R. x~G It follows immediately that Q c~R =P c~ R. The right ideal (P c~R)U of U is actually two-sided and the_ring U:=U/(Pc~R)U carries the structure of a crossed product [7~-/~ [G], R." =RIP c~ R (1.2) (d). If U is simple, we have P=Q=(Pc~R)U. Otherwise, by Lemma(1.6), there exists x E G, x 4= 1, such that ~(x) is an inner automorphism of/~. Since G is simple it follows from (1.2)(b) that each ~(x) is inner. Hence for each xEG there exists a unit rx~R such that Zx:=r~2~U centralizes /~. Let C:=centre(/~), a field. Since U= ~ /~z~ we have Cacentre([7) and we may consider U as a C-algebra. x~G We have Zx z~ =(rx ~)(r. 9): rx r; (x} ' ~(x, y)~= rx~ {x)-' ?(x, y) rL 1 zx.=c~.y z.,, where Cx, y~ C. Hence A." = ~ C zx is a C-subalgebra of [7 and obviously U = R.A. x~G Since A centralizes the central-simple subalgebra/~the algebra [7 has the structure of a tensor product, [7 =/~ @ A, and the ideals of U are of the form/~ @ I, where I C C is an ideal of A ([9], p. 110). Obviously / has to be prime if/~ @ I is prime. Since A C is a finite-dimensional algebra, we conclude that all prime ideals in U are maximal. In particular P = Q. This finishes the proof of the theorem. (1.8) Proposition. Let R be a ring, G a finite group and U:=R~ [G] a crossed product. (a) If U is prime and R is primitive, then U is primitive. (b) If R is prime and U is primitive, then R is primitive. Proof (a). Let V be a faithful irreducible R-module. Then W--= V@ U~Mod- U R has finite length and its annihilator is ( (-] annR(V) ~(~)) U =(0). Fix a composition x~G series W=WomWlm...mW,=(O) of W and let Qi:=annv(~_x/~). Then QaQ2 ... Q~cannv(W)=(O). Since U is prime it follows that Qi=(0), some i. Hence U is primitive. (b) Let VeMod- U be faithful and irreducible. By (1.3) we have 290 M. Lorenz VR~-- ~ @W~, W~Mod-R irreducible. If Qg denotes the annihilator of W~, i=1 then (~ Qi = annR (V) = (0) and since R is prime, Qi = (0) some i. Hence R is primitive. i=1 (1.9) Remarks. The primitivity of R does not in general imply R~ [G] to be prime as can be seen for instance by considering the group algebra of a finite group over a field. Conversely, the primitivity of R~ [G] does not imply the primeness of R: Let R=Y| ~f a field. Let yeAut(R) be given by (2, #)Y=(#, )0 (2, #~f) and let U,= R~ [7/2] be the corresponding crossed product of R over ~2 = (Y). Then U is simple, hence primitive, but R is not prime. 2. Fixed Rings (2.1) Notations and Conventions. In the following R will always mean a ring with unit dement, G a finite group of automorphisms of R and R ~ the fixed ring, RG= {reRlrX=r for all x~G}. Furthermore U:=R~ [G] will denote the crossed product of G over R with trivial factor set ~-1 and ~ given by the embedding of G into Aut (R). For convenience, we shall assume IG[ is invertible in R throughout the re- mainder of this paper. An easy consequence of this assumption is the following: Let I be a G-stable ideal of R, let/~:= R/I and let G ~ Aut (/~) be induced by G. Then /~o=/~e (see [8], p. 11). We set e,=lG[ -1 ~ 2EU. Then e is an idem- potent of U. x~o (2.2) R is a left U-module via the action (~ rx 2) r~ = ~ rx r ~ 2. We have a U-iso- morphism uR_~Ue given by r--~r ~ ~ and a ring isomorphism RG~-EndvR xEG associating to rER ~ the right multiplication by r ([7], Lemma 1.2). Therefore RV~-e Ue. (2.3) Theorem (Bergman:Isaacs [-4]). (i) If R G is nilpotent, then R is nilpotent. (ii) If R is semiprime, then R a is semiprime. (iii) If I is a G-invariant right ideal of R, then either I is nilpotent or 1 ~ R a 4: (0). (Here we do not assume that R has a 1.) (2.4) Lemma. (a) Let VaMod R be of finite length. Then the restricted module VR~ is of finite length. (b) If V is completely reducible, then VR~ is completely reducible. Proof. (1) Let A be an arbitrary ring, f =f 2 eA and WeMod-A. If W is of finite length, then so is WfeMod-fAr. If W is completely reducible, then the same holds for Wf : Let X c Wf be an fA f-submodule of Wf. Then XA =X| There- fore, if X 1 ~X 2 are fAf-submodules of Wf, we obtain X 1 A ~X 2 A. The first Crossed Products and Rings with Finite Group Actions 291 assertion follows. As to the second, notice that by hypothesis there are A-homo- morphisms ~z,#:XA ~ W suchthat ~z#=idxA. Let #1:=#Ix and 7Cl'.=TC[w:. Then #1: X-, Wf and ~h: Wf~X are fAf-linear and satisfy 7h #1 = idx- (2) One has eUe=R~ e: Let rsR and yeG. Then e(ry)e=er(ye)=ere=(lG[ -1 ~ "2r)e=lG[ -1 ~ r ~ ~e=(]G] -1 ~ r~-~)e. xEG x~G x~G Since IG[-l~r ~ I~RG we have eUecR6e. As to the other inclusion it suffices x~G to remark that for r e R G we have r e = e r = e r e. (3) Consider the module V:=(V@U)e~Mod-eUe. Direct calculation R shows that V={ ~ v| Hence we have an RG-isomorphism ~: V~V, x~G (v)= ~ v| By step (2), the e Ue-submodules of l/ coincide with the R G- x~G submodules of I7. The assertions of the lemma now follow from step (1), because V of finite length certainly implies V@ UcMod-U to be of finite length, and if R V is completely reducible, then V@ U is completely reducible, by (1.4). R (2.5) Corollary (Montgomery [10]). J(R ~) = J(R) c~ R ~. Pro@ The inclusion J (R ) c~ R ~ ~ J ( R ~) follows readily from the fact that U (R ) c~ R a = U(R~).-Let x~J(R ~) and let V~Mod-R be irreducible. By the lemma, VRG is completely reducible. Therefore Vx = (0), and x sJ(R). (2.6) The following result has been obtained independently by Reiter [12]. Corollary. (a) soc(R) c~ R ~ c soc(Ra). (b) Equality holds if R is semiprime. Proof. Let xesoc(R) c~R ~. Then the right ideal xR of R is completely reducible. Using the lemma we conclude that x RGc X R[R~ is completely reducible. There- fore x ~soc(Ra). - The second assertion follows from Fisher-Osterburg [7], Lemma 1.9. (2.7) Theorem. All primitive ideals of R are maximal if and only if the primitive ideals of R ~ are maximal. Proof. If the primitive ideals of R are maximal, then by Theorem (1.7), the same holds in U. Now RG-----eU e (2.2). Therefore (see I-9], p. 206) Priv (R G) is homeo- morphic to a subspace of Priv(U) and hence inherits the property ~, which means that the primitive ideals of R a are maximal. 292 M. Lorenz In order to prove the other implication, fix a primitive ideal P of R and an irreducible R-module V such that P = ann~ (V). Let fi: = 0 px and /~: = RIP. Then V is an irreducible /~-module such that xEG annRe (V) = P/P c~ R G= (0). By (2.4), V is completely reducible of finite length when considered as an RG-module. Therefore (0)= annRe(V)= P~ for some primitive i=1 ideals P~ of/~. As/~ is a homomorphic image of R e, all primitive ideals of R ~ are maximal. Using the Chinese remainder theorem we conclude that R a is a finite direct sum of simple rings. By Fisher and Osterburg ([8], Theorem 4.3), /~ is a finite direct sum of simple rings. This forces the prime ideal p//5 of/~ to be maximal, as was to be shown. (2.8) Proposition. (a) Let R be prime. Then R is primitive if R G is primitive. (b) If R is primitive and R e is prime, then R a is primitive. (c) If R G is primitive and U is prime, then U is primitive. (d) Suppose U is prime. Then the following implications hold: R primitive ~ U primitive ~ R e primitive. Proof. (a) Let VcMod-R G be faithful and irreducible. We shall construct an irreducible R-module W containing V: For this write V ~- Re/I, I a maximal right ideal of R e. Now IR ~= R, for otherwise i I r~ = 1 (i~I, rLeR ) would imply l [G]= ~ I x= 2 (~,iIrl) ~= ~ ~il~ix=~it(2 ~)~I x~O x~G l x~G t l x~e and hence lsI since [G[ is invertible. Therefore we can find a maximal right ideal J of R containing IR. The module W: = R/J~Mod- R is irreducible and contains R G+ J/J = Re/R Gc~ J =Re/I = V. Let P be its annihilator in R. Then Pc~RacannR~(V), which is zero, since V is faithful. Therefore, by the Bergman-Isaacs' theorem (2.3), we conclude that (-] W=(0). Finally, the primeness of R yields P=(0). Thus R is primitive. x~G (b) R has a faithful irreducible module V. By (2.4) VR~ is completely reducible of finite length: VR~ = V1 | O V~, Vi ~ Mod - R G irreducible. Writing Qi:=annR~(Vi) we obtain (~ Qi=annR~(V)=(O ). i=l Using the assumption on R G we conclude that Q~=(0), some i, and hence R G is primitive. (c) We have R~-e Ue (2.2). Since e Ue is primitive, there exists a primitive ideal P of U such that ePe=(0) (see [9], p. 206). Again, the primeness of U yields P=(0). (d) The first implication is contained in (1.8), and the second again follows immediately from the fact that RG~ e U e ([91 1.c.) Crossed Products and Rings with Finite Group Actions 293 (2.9) Remarks. The following examples (taken from [7], [-8]) show that the prime- ness assumptions in Proposition (2.8) are not superfluous. Let R:=M2(d), the 2 x 2-matrix ring over the field ~ of characteristic +2, and let g~Aut(R) be given by g (~ bd)=(a c db).ThenRiscertainlyprimi - tire, whereas R References L Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Berlin, Heidelberg, New York: Springer 1974 2. Armendariz, E.P.: On fixed rings of automorphisms. Notices Amer. math. Soc. 24, A-57 (1977) 3. Azumaya, G.: Galois theory for uniseriat rings. J. math. Soc. Japan 11 130-146 (1949) 4. Bergman, G.M., Isaaes, f.M.: Rings with fixed-point-free group actions. Proc. London math. Soc., III. Ser. 27, 69-87 (1973) 5. Bovdi, A.A. : Crossed products ofa semigroup and a ring. Soviet Math., Doklady 2, 438-440 (1961) 6. Dixmier, J.: Repr6sentations irrdductibles des algdbres de Lie nilpotentes. Anais Acad. Brasil. Ci. 35, 491-519 (1963) 7. Fisher, J.W., Osterburg, J.: Some results on rings with finite group actions. In: Ring Theory (Athens, Ohio 1976) pp. 95-111. Lecture Notes in Pure and Applied Mathematics. Vol. 25. New York: Dekker 1977 8. Fisher, J.W., Osterburg, J.: Semiprime ideals in rings with finite group actions. J. Algebra-to appear 9. Jacobson, N.: Structure of Rings. American Mathematical Society Colloquium Publications. Vol. 37, Rev. Ed. Providence: American Mathematical Society 1964 10. Montgomery, S.: The Jacobson radical and fixed rings of automorphisms. Commun. Algebra 4, 459-465 (1976) 294 M. Lorenz 11. Passman, D.S.: The Algebraic Structure of Group Rings. New York-London: Wiley-Interscience 1977 12. Reiter, E.E.: Finite group actions on rings. Ph. D. thesis, University of Cincinnati, Cincinnati, Ohio, in preparation 13. Snider, R.L.: Primitive ideals in group rings of polycyclic groups. Proc. Amer. math. Soc. 57, 8-10 (1976) 14. Zalesskij, A.E.: Irreducible representations of finitely generated nilpotent torsion-free groups. Math. Notes 9, 117-123 (1971) Received July 21, i977 Note Added in Proof Prof. Fisher kindly pointed out to me that from this article it follows easily that ([Gd-leR) V~Mod-R noetherian (artinian)~ VR~ noetherian (artinian), V has Krull dimension ~ VR~ has Krull dimension and K dim VR = K dim VRG.