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Primitive Ideals in Crossed Products and Rings with Finite Group Actions

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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 ~..
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