Fair-sized projective modules

Fair-Sized Projective Modules

PAVELPRÏIÂHODA(*)

ABSTRACT- We investigate a condition on particular chains of ideals that allows us to determine properties of infinitely generated modules over noetherian rings. The results apply to semilocal noetherian rings, integral group rings of finite groups and universal enveloping algebras of solvable Lie algebras of finite dimension.

1. Introduction.

This paper is devoted to the study of infinitely generated projective modules over associative unitary rings. We are interested in the case in which the ring has projective modules that are not direct sums of finitely generated modules. Some general results and examples of rings with such modules were given in [12]. Our motivation was to find a technique that could be applied to prove the existence of superdecomposable projective modules over semilocal rings.

Let us briefly explain the main idea of the paper. According to a well known theorem of Kaplansky, any projective right module over a ringRis a direct sum of countably generated right modules, so it suffices to in- vestigate countably generated projectives, that is, direct summands of a countably generated free right moduleFˆR^(N)_R . Suppose thatPP⁰ˆF.

The canonical projection p:F!P is given by a column-finite NN idempotent matrix A. We say that ArepresentsP(observe that the col- umns ofAgenerateP). LetInbe the ideal ofRgenerated by the entries of Athat are below then-th row. Clearly,Pis finitely generated if and only if there exists k2N such that I_lˆ0 for every lk. The other possible extreme case is whenI₁ˆI₂ˆ ˆR. It is a well-known result of Bass

(*) Indirizzo dell'A.: Charles University in Prague, Faculty of Mathematics and Physics, Department of Algebra, Sokolovska 83, 186 75 Prague, Czech Republic.

E-mail: paya@matfyz.cz

Supported by Eduard CÏech center for algebra and geometry, grant LC505.

[3, Theorem 3.1] that in this case P'F provided R=J(R) is right noe- therian. In this paper, we focus our attention on the case in which the sequenceI₁I₂. . .terminates at an idealI. It is easy to see thatIis idempotent. We show that if R is left and right noetherian and the se- quenceI₁I₂ terminates atI, thenPcontains as a direct summand any countably generated projective module having its trace ideal in I.

Cf. [3, Theorem 3.1]. The following easy condition assures that any se- quence of ideals derived from an idempotent column-finite matrix termi- nates: IfI₁;I₂;. . .is a sequence of ideals in Rsuch thatI_k‡1I_kˆI_k‡1 for any k1, then there exists n such that I_nˆI_n‡1ˆ. . .: Call (*) this condition.

In section 2, we show that over a left and right noetherian ringRsa- tisfying condition (*), the theory of projective modules ``reduces'' to the theory of idempotent ideals inRand the theory of finitely generated pro- jective modules over the factor rings ofRmodulo idempotent ideals. This explains and is related to the statement in the introduction of [3], according to which ``infinitely generated projective modules invite little interest''.

The remaining sections are devoted to presenting some examples. We prove that (*) holds for semilocal noetherian rings, integral group rings of a finite group and universal enveloping algebras of finite solvable Lie alge- bras over a field of characteristic zero. This allows us to prove that:

(i) There exists a semilocal noetherian ring with superdecomposable projective modules.

(ii) Indecomposable projective modules over integral group rings of finite groups are finitely generated.

(iii) Any infinitely generated projective module over a finite dimen- sional solvable Lie algebra over a field of characteristic zero is free.

Notice that (ii) solves [9, Problem 8.34].

Let us briefly recall some notions and fix the notation. The word ``ring'' always means associative ring with an identity and ``module'' means unital right module. IfMis a module overR, then P

f2HomR(M;R)f(M) is an ideal ofR called thetrace ideal of M. We denote it Tr(M). IfPis a projective module overR, then Tr(P) is the smallest idealXofRsuch thatPXˆP, and is an idempotent ideal. Further ifXis a subset of a ringR, we denoteRXRthe (two-sided) ideal generated by X. In case Xˆ fxg we denote RxR the smallest ideal of R containing x. Notice that in general the relation RxRˆ frxsjr;s2Rgdoes not hold. Recall the following important result due to Whitehead:

FACT1.1 [18, Corollary 2.7]. Let I be an idempotent ideal of R finitely generated on the left. Then there exists a countably generated projective right R-module P such thatTr(P)ˆI.

To avoid confusion, we will call the rings which have all left ideals and all right ideals finitely generated left and right noetherian rings, although they are often called noetherian rings. Finally, we will call infinitely generated projective modules the projective modules that are countably generated but not finitely generated.

2. I-big modules

LetPbe a countably generated projective module over a ringRand let Ibe an ideal of R. We say thatPisI-bigif for any countably generated projective module Q with trace ideal contained in Ithere exists an epi- morphism ofPontoQ. Hence, in this case,Pcontains a direct summand isomorphic to Q. Notice that this definition will be applied to countably generated projective modules only.

REMARK2.1 (Eilenberg's trick). LetIbe an ideal of a ringRand letP be an I-big projective module. If Q is a countably generated projective module with trace ideal contained inI, thenPQ'P, becauseQ^(v)is a direct summand ofP.

LEMMA 2.2. Let I be an idempotent ideal that is finitely generated as a left ideal. Then there exists an I-big projective module P such that Tr(P)ˆI. Such a module P is unique up to isomorphism.

PROOF. By [18, Corollary 2.7], there exists a countably generated projective module P with Tr(P)ˆI. Clearly, Tr(P^(v))ˆI. If Q is a countably generated projective module having the trace ideal contained in I, thenQIˆQandQis a factor of P^(v). LetP1;P2 beI-big modules such that Tr(P1)ˆTr(P2)ˆI. By Remark 2.1, P1P2'P1. Similarly,

P₁P₂'P₂. ThusP₁'P₂. p

REMARK2.3. We have just proved that, for any idealIthat is a trace ideal of a countably generated projective module, there exists a unique countably generated projective moduleP(up to isomorphism) such thatP isI-big and Tr(P)ˆI. We will make use of I-big modules over left and

right noetherian rings. Observe thatR^(v)is anR-big projective module and that anyR-big projective module has trace idealR. Therefore anyR-big projective module is isomorphic toR^(v).

We say that a ringRsatisfies Condition (*) if for any sequenceI1;I2;. . . of ideals inRsuch thatI_k‡1I_kˆI_k‡1;k2Nthere existsn2Nsuch that I_kˆIn for any nk2N. Notice that such a sequence is necessarily a descending chain.

We will use this condition in the following context: Suppose we have a countably generated projective moduleP. ThusPis a direct summand of a countably generated free module,PP⁰ˆR^(N) say. The canonical pro- jectionp:R^(N)!P can be written with respect to the canonical basis of R^(N)as anNNmatrixAˆ(a_i;j)_i;j2Nwith entries inR. Moreover,Ais a column-finite matrix (that is, for anyj2Nthere existsi2Nwitha_k;jˆ0 wheneverki). ThereforeA²is defined and it is easy to see thatA²ˆA (that is, A is an idempotent matrix). Conversely, given any idempotent column-finite NN matrix A, the corresponding module PˆAR^(N) is projective.

Now, letAˆ(a_i;j)_i;j2N be an idempotent column-finite matrix over R and letI_k ˆ P

ki2N;j2NRa_i;jR; k2N. For anyk2Nthere exists an integer n_k>ksuch thata_i;jˆ0 wheneverin_kandj5k. SinceAis idempotent, we have I_nkI_k ˆI_nk. Hence there exist positive integers m₁5m₂5 such that Imj‡1Imj ˆImj‡1. IfR satisfies (*), then there existsl2Nsuch thatImj ˆIml for anylj2N, in particular,Imj ˆImj‡1ˆImj‡1Imj ˆI²_mj ifjl. So ifIˆ \_j2NI_nj, thenIis an idempotent ideal andI_jˆIfor almost allj2N.

We will say that a projective modulePover a ringRisfair-sizedifPis countably generated and the set I(P):ˆ fIjIis an ideal ofR such that P=PI is finitely generatedg has a least element. The following lemma shows that any countably generated projective module over a ring sa- tisfying (*) is fair-sized. Moreover, the proof reveals the relation between the smallest element ofI(P) and an idempotent matrix representingP.

LEMMA 2.4. Let R be a ring satisfying(*)and let P be a countably gen- erated projective module over R. The setfIjI is an ideal of R such that P=PI is finitely generatedghas a least element I₀, which is an idempotent ideal.

PROOF. Let Aˆ(a_i;j)_i;j2N be an idempotent column-finite matrix representing P, and I_k;k2N, be the ideals defined above. Set

I0ˆ \k2NIk. As remarked above,I0 is idempotent. Let feiji2Ng be the canonical free basis of R^(N) and suppose that I0ˆImˆIm‡1ˆ. . . Then ^mP¹

iˆ1 Ae_iR‡PI₀ˆP, so P=PI₀ is finitely generated. Let K be an ideal such that P=PK is a finitely generated module. Assume PˆAR^(N) R^(N). Notice that PKˆP\K^(N), that is, the elements of PK are exactly the elements of P having all their components inK. If P=PK is finitely generated, then there exists k2N such that a_i;j2K

for everyikand j2N. ThereforeI₀K. p

Thus ifRsatisfies (*), every countably generated projective moduleP determines a pair (I;P⁰), whereIis an idempotent ideal andP⁰is a finitely generated projective module over R=I. More precisely, Iis the smallest ideal ofRsuch thatP=PIis finitely generated andP⁰is the moduleP=PI considered as anR=I-module in the obvious way. IfPis a countably gen- erated projective module, then the corresponding idempotent ideal I is given by a matrix representingPasIˆ \k2NIk, but the characterization of Iin Lemma 2.4 implies thatIis independent of the choice of the matrix (and of the complementP⁰in the decompositionPP⁰ˆR^(N)).

LEMMA 2.5. Let I be an idempotent ideal of a ring R such that I is fi- nitely generated as a left module and as a right module. If P and Q are I-big projective modules satisfying P=PI'Q=QI, then P'Q.

PROOF. Let B be the unique I-big projective module having trace idealI. Observe thatPB^(v)'Pby Remark 2.1. Iff:P!Qinduces an isomorphism P=PI!Q=QI, then f(P)‡QIˆQ. Since QI is countably generated and Tr(B)ˆI, we get an epimorphismh:PB^(v)!Qsuch that hj_Pˆf. Asfinduces a monomorphismP=PI!Q=QIandh(B^(v))QI, we getXˆKerhPIB^(v). ThusX is a direct summand ofPIB^(v). In particular,XIˆX. Consequently, Tr(X)I, soQX'Qby Remark 2.1.

Finally,Q'QX'PB^(v)'P, andQ'Pfollows. p The following lemma is a straightforward extension of [18, Corollary 2.7].

LEMMA 2.6. Let I be a proper idempotent ideal of a ring R. Assume I finitely generated as a left ideal. Let P⁰be a finitely generated projective module over R=I. Then there exists an I-big projective module P such that P=PI'P⁰.

PROOF. We will find a countably generated projective moduleP0such thatP0=P0I'P⁰.

Suppose thatP⁰is given by annnmatrixXidempotent moduloI.

The R-matrix X is a lifting of an idempotent R=I-matrix X. Let IˆIi₁‡ ‡Ii_l;i₁;. . .;i_l2I. Construct a sequence of matrices

A1;A2;. . . as follows: A1 has c1ˆn columns and r1ˆln‡n rows. The

square matrix given by the first n rows of A1 is X, (A1)_i;j ˆ0 if n5in‡(j 1)lor i>n‡jl, and the remaining entries in each col- umn are filled with the generatorsi₁;. . .;i_l. That is, the matrixA₁written in blocks is

X

b 0 0 0

0 b 0 0

0 0 0 ... b 0

BB BB BB

@

1 CC CC CC A

;

wherebis the column (i₁;. . .;i_l)^T.

If A_k;r_k;c_k have been defined, then A_k‡1 has c_k‡1ˆr_k columns and rk‡1ˆrk‡lrk rows. The nn top left corner of Ak‡1 is given by the matrixXand all the other entries in the firstr_krows ofA_k‡1are zero. The remaining lr_k rows contains i₁;. . .;i_l placed in each column in the same

``independent manner'' as described forA₁.

We claim that for anyk2Nthere is ac_k‡1r_k‡1matrixB_k such that BkAk‡1Ak ˆAk. Observe that theckckmatrix given by the firstckrows ofA_k is idempotent moduloI. We can find anr_kr_k matrixC_k such that C_kA_kˆA_k: Thenntop left corner ofC_kis given byX, the other entries in the first c_k columns are zero and the matrix C_k can be completed by elements of I because IˆIi₁‡ ‡Ii_l and i₁;. . .;i_l are placed in- dependently in the bottom part ofAk. This matrix Ck can be written as DkAk‡1, whereDk is a suitablerkrk‡1 matrix. (Again we placeXin the top left corner ofD_k, and put all the other entries in the firstr_kcolumns of D_k equal to zero. The remaining entries can be completed because the generators of I are placed independently in A_k‡1.) Now, since AkˆCkAkˆDkAk‡1Ak, putBkˆDk.

View the free moduleFkˆR^ck as the set of columns of lengthck. Let f_k:F_k!F_k‡1 be the homomorphism given by f_k(u)ˆA_ku for every u2F_k. By [18, Theorem 2.1], the colimit of the direct system induced by thef_k's is a projective moduleP₀. Obviously, P₀is a countably generated module. Applying the functor _RR=I:Mod-R!Mod-R=I, we see that

P0=P0I is an R=I-module isomorphic to the colimit of the system (R=I)ⁿ !^X (R=I)ⁿ!^X , which is easily seen to beX(R=I)ⁿ'P⁰. Therefore P₀=P₀I'P⁰.

Finally, by Lemma 2.2, there exists anI-big projective moduleBsuch that Tr(B)ˆI. Since BIˆI, P:ˆP0B is an I-big projective module

withP=PI'P0=P0I'P⁰. p

REMARK2.7. Let us explain the construction in the proof of Lemma 2.6 via an example. Suppose thatIis a proper idempotent ideal of a ringR such thatIˆIi1‡Ii2for somei1;i22I. Letx2Rbe such thatx‡Iis an idempotent element ofR=I, i.e.,x x²2I. Then there aret1;t22Isuch that xˆx²‡t₁i₁‡t₂i₂. Further, there are u₁;u₂;v₁;v₂2I such that u₁i₁‡u₂i₂ ˆi₁ andv₁i₁‡v₂i₂ˆi₂. Set

A₁ˆ x i₁ i₂ 0

@ 1

A C₁ˆ x t₁ t₂ 0 u₁ u₂ 0 v₁ v₂ 0

@

1

A C₁⁰ ˆ x x² t₁ t₂ 0 u₁ u₂ 0 v₁ v₂ 0

@

1 A:

Obviously,C1A1ˆA1. Moreover, all entries ofC₁⁰are inI. Therefore there is a 36 matrixTˆ(t_i;j)_1i3;1j6satisfyingTA⁰₂ˆC₁⁰, where

A⁰₂ˆ i₁ i₂ 0 0 0 0 0 0 i₁ i₂ 0 0 0 0 0 0 i₁ i₂ 0

@

1 A

T

:

All the entries ofTcan be chosen inI, but this is not important. It is easy to see thatC1ˆB1A2, where

B₁ˆ x 0 0 t_1;1 t_1;2 t_1;3 t_1;4 t_1;5 t_1;6 0 0 0 t_2;1 t_2;2 t_2;3 t_2;4 t_2;5 t_2;6 0 0 0 t_3;1 t_3;2 t_3;3 t_3;4 t_3;5 t_3;6 0

@

1 A

A₂ ˆ x 0 0 i₁ i₂ 0 0 0 0 0 0 0 0 0 i₁ i₂ 0 0 0 0 0 0 0 0 0 i1 i2

0

@

1 A

T

:

The following lemma is, in a sense, a restatement of [3, Theorem 3.1].

We prefer to give a brief but complete proof of the statement for left and right noetherian rings rather than specifying what should be modified in the proof of [3, Theorem 3.1] to get a real generalization.

LEMMA 2.8. Let R be a left and right noetherian ring. Let Aˆ(ai;j)_i;j2N be an idempotent column-finite matrix. Set Ikˆ P

ik;j2NRai;jR. If there exists n02N such that ImˆIn0 for every mn0, then the module PˆAR^(N)R^(N)is In0-big.

PROOF. Set IˆIn0 and observe thatI is finitely generated as a left ideal. Leta_ibe thei-th column ofA. We will prove the following claim. For any n2N there exist m2N and r₁;. . .;r_m2R such that if a₁r₁‡ ‡a_mr_mˆ(c_i)_i2N, thenIP

inRc_i. By induction, define positive integerss₁;. . .;s_k,s⁰₁;. . .;s⁰_k andx₁;. . .;x_k2Rsuch that P^l

iˆ1Rx_i4 ^l‡1P

iˆ1Rx_i for every 1l5kandIP^k

iˆ1Rx_i.

Puts1ˆ1,s⁰₁ˆnandx1ˆa_s⁰_1;s1. IfIRx1, we have finished. Other- wise, suppose we have defined positive integers s₁;. . .;s_l, s⁰₁;. . .;s⁰_l and x1;. . .;x_l2R such that I6P^l

iˆ1Rx_i. Since R is right noetherian, there exists m_l2N such thatm_l>s_l and P

j2Na_s⁰_l;jRˆ P

1j5ml

a_s⁰_l;jR. Since Ais column-finite, there exists m⁰_l2N with m⁰_l>s⁰_l and ai;jˆ0 whenever im⁰_landjm_l. AsII_m⁰_l, there exists⁰_l‡1>m⁰_l,s_l‡1>m_landt_l‡12R such thata_s⁰_l‡1;sl‡1t_l‡162P^l

iˆ1Rx_i. Putx_l‡1ˆa_s⁰_l‡1;sl‡1t_l‡1.

SinceRis left noetherian, this process must stop, that is, there existsk such thatI P

1ikRx_i. It follows that there arer1;. . .;rsk 2Rsuch that thes⁰_i-th component of P^sk

iˆ1airiisxi for any 1ik. This is obvious for kˆ1. If k>1, note that s15m15s25m25 5mk 15sk and s⁰₁5m⁰₁5s⁰₂5m⁰₂5 5m⁰_{k 1}5s⁰_k. Further, P

j2Nas⁰₁;jRˆ^mP^{1 1}

jˆ1 as⁰₁;jR

and P

j2Na_s⁰_i;jRˆ ^mP^{i 1}

jˆmi1

a_s⁰_i;jR if 2i5k. Moreover, a_i;j ˆ0 for any 1jm_landim⁰_l. This concludes the proof of the claim.

Now we can construct a sequence p₁;p₂;. . . of elements in P, p_iˆ(c_j;i)_j2N say, such that there exist integers 1ˆi₁5i₂5 with IRcik;k‡ ‡Rcik‡1 1;kfor anyk2Nandcl;k ˆ0 for anylik‡1. W e proceed by induction again. Puti1ˆ1. By the claim, there existsp1such that I P

j2NRc_j;1. Of course, there exists i2>i1 with c_l;1ˆ0 for every li₂.

Suppose we havep₁;. . .;p_k andi₁;. . .;i_k‡1. By the claim, there exists

pk‡1 such that I P

jik‡1

Rcj;k‡1. Let ik‡2>ik‡1 be an integer such that c_j;k‡1ˆ0 for allji_k‡2.

Now, letQbe a countably generated projective module with trace ideal contained inIgiven by a column-finite idempotent matrixBoverR(again, we considerQas a submodule ofR^(N)). Since the trace ideal ofQlies inI, all entries ofBare inI. LetCbe a matrix such that columns ofCare given by

p₁;p₂;. . .: The shape of C guarantees the existence of a column-finite

matrixDhaving all entries in Tr(Q) such thatDCˆB(it is important to realize that the elements ofDcan be chosen inI). Now, letf:R^(N)!R^(N) be given by D. Observe, that Qf(P) and that ifp:R^(N)!Q is a pro- jection, thenpfj_Pis an epimorphism ofPontoQ. HencePisI-big. p REMARK2.9. Imitating the proof of [3, Theorem 3.1], we could get the following. LetRbe a ring such thatR=J(R) is right noetherian. LetP;I_kbe as above and suppose thatIˆI_n ˆI_n‡1ˆ is a finitely generated left ideal such thatI\J(R)ˆJ(R)I. ThenPisI-big. (ForIˆRwe get Bass' big projectives theorem). Also we could omit the assumption (*) and prove that Pis\n2NIn-big. We do not give the details because we do not have applications for this version of Lemma 2.8.

Comparing the definition of I0 in the proof of Lemma 2.4 and the statement of Lemma 2.8, we immediately get

COROLLARY2.10. LetRbe a left and right noetherian ring satisfying (*). IfP is a countably generated projectiveR-module andI is the least ideal ofRsuch thatP=PIis finitely generated, thenP isI-big.

Obviously, Lemma 2.8 can be applied to study projective (right) mod- ules over left and right noetherian rings satisfying (*). The following lemma shows that over these rings we can apply Lemma 2.8 also for pro- jective left modules. Recall that anNNmatrixAˆ(a_i;j)_i;j2N is said to berow-finiteif for anyi2Nthere existsj2Nsuch thata_i;kˆ0 for every kj.

LEMMA 2.11. Let R be a left and right noetherian ring satisfying (*).

Let Aˆ(a_i;j)_i;j2Nbe a row-finite matrix over R such that A²ˆA. For any k2Nlet I_kˆ P

j>k;i2NRa_i;jR. Then there exists n2Nsuch that I_mˆI_nfor any mn.

PROOF. Throughout the proof, we will work inside the left module FˆRR^(N). Lete1;e2;. . .be the canonical free basis ofF. For anyi2N, let a_ibe thei-th row ofA, that is,a_iˆ(a_i;1;a_i;2;. . .)2F. ThusAgives a left projective modulePˆFAˆ P

i2NRa_i. For anyl2N₀letp_l:F!_RR^lbe the projection given byp_l((x₁;x₂;. . .))ˆ(x₁;. . .;x_l) (as usual,_RR⁰is the trivial leftR-module). For anyi2N letc_i:F!_RRbe the projection given by c_i((x₁;x₂;. . .))ˆx_i.

Construct integers 0ˆn15n25 and ideals J1J2 as fol- lows: Putn1ˆ0 andJ1ˆ P

i;j2NRa_i;jR. Suppose thatn_k andJ_k have been defined. SinceRis left noetherian, there existsl2Nsuch that the module pnk(P) is generated bypnk(a1);. . .;pnk(a_l). AsAis row-finite, there exists m>n_k such that a_i;m⁰ ˆ0 for any 1il;m⁰m. Set n_k‡1ˆm. Let J_k‡1 be the ideal generated byfr2Rjthere existp2Pandi2N such thatp_nk‡1(p)ˆ0 andc_i(p)ˆrg. We claim thatJ_k‡1J_kˆJ_k‡1. In order to prove the claim, it suffices to prove thatSJk‡1Jk for a setSgenerating J_k‡1. Letp2Pbe such thatpnk‡1(p)ˆ0. Write pˆ(0;. . .;0;rnk‡1‡1;. . .).

Then pˆr_nk‡1‡1(e_nk‡1‡1A)‡r_nk‡1‡2(e_nk‡1‡2A)‡ From the construction it follows that for anyi2Nthere existsp_i2P such thatp_nk(p_i)ˆ0 and c_nk‡1‡j(p_i)ˆc_nk‡1‡j(e_nk‡1‡iA) for every j2N. Since c_nk‡1‡j(p_i)2J_k, the equation

r_nk‡1‡iˆc_nk‡1‡i(p)ˆrnk‡1‡1c_nk‡1‡i((enk‡1‡1)A)‡rnk‡1‡2c_nk‡1‡i((enk‡1‡2)A)‡ implies thatJ_k‡1ˆJ_k‡1J_k.

As R satisfies (*), there exists m2N such that JmˆJm‡1ˆ Clearly,JkInk for anyk2N. On the other hand,Ink‡1Jnk. This con-

cludes the proof of the lemma. p

Let R be a ring, let Vr(R) be a set of representatives of the finitely generated projective rightR-modules,V_l(R) be a set of representatives of the finitely generated projective left R-modules, V_r(R) be a set of re- presentatives of the countably generated projective right modules and V_l(R)be a set of representatives of the countably generated projective left R-modules. In the following theorem we considerVr(R=R) andVl(R=R) as sets containing one element.

THEOREM2.12. Let R be a left and right noetherian ring satisfying(*).

LetId(R)be the set of its idempotent ideals and letSbe the disjoint union [__I2Id(R)Vr(R=I). Then there is a bijectionW:Vr(R)! S. Moreover, there exists a bijection between Vr(R)and V_l(R)extending the classical bijec- tion between V_r(R)and V_l(R)induced byHom_R( ;R_R).

PROOF. By Corollary 2.10, any countably generated projective right module P isI-big, where I is the least ideal such that P=PI is finitely generated. We know thatIis idempotent. This gives a map ofVr(R)intoS.

This map is a bijection by Lemmas 2.5 and 2.6.

Of course, all the results of this section can be formulated for left modules. We do not know whether condition (*) is equivalent to condi- tion (*'): LetI1;I2;. . .be a sequence of ideals such that IkIk‡1 ˆIk‡1 for anyk2N. Then there existsn2Nsuch thatIn ˆIn‡1ˆ. . .Condition (*) is connected to right modules while (*') is connected to left modules.

Therefore it would be more precise to talk about condition right (*) instead of (*). In order to be concise, we have omitted the word ``right'', but the reader should be aware that this condition has to be changed formulating the versions of the results for left modules. However, we can use Lemma 2.11 and the ``left version'' of Lemma 2.8 to see that any countably gen- erated projective left moduleQisI-big, whereIis the least ideal such that Q=IQis finitely generated. Again,Iis idempotent and finitely generated as a right module, therefore the ``left versions'' of Lemma 2.5 and Lemma 2.6 give a bijection of V_l(R) and the disjoint union[__I2Id(R)V_l(R=I). The bi- jection betweenV_r(R) andV_l(R), then follows from the dualities between finitely generated projective left and rightR=I-modules, whereIvaries in

Id(R). p

REMARK 2.13. Observe that if R is a left and right noetherian ring having (*), then every indecomposable projective module is finitely gen- erated. Although we think that (*) is a very particular property (see [8] for examples of rings having infinite properly descending chains of idempo- tent ideals), it seems to occur quite often in natural examples of left and right noetherian rings.

3. Semilocal noetherian rings.

Recall that a ringRis said to besemilocal, if the factor ofRmodulo its Jacobson radical is semisimple artinian. Throughout the paper, J(R) de- notes the Jacobson radical of R. If P;Q are projective modules, then P=PJ(R)'Q=QJ(R) if and only ifP'Q[13, Theorem 1.3]. In this section, we show that semilocal left and right noetherian rings satisfy (*), so that any countably generated projective module over such a ring is fair-sized.

Further, we show a connection between the pair (I;P=PI) defined in the previous section and the semisimple moduleP=PJ(R). Finally, we give an

example of superdecomposable projective module over a semilocal noe- therian ring.

Recall that ifPis a projective module overR, then the intersection of all maximal submodules ofP, called theradicalofP, is rad(P)ˆPJ(R). IfRis semilocal andS₁;. . .;S_kare representatives of the simpleR-modules (that is, for any simpleR-moduleSthere exists exactly onei2 f1;. . .;kgwith S'Si), then for every projective modulePthere are cardinalsl1;. . .;lk, uniquely determined, such thatP=PJ(R)ˆS^(l₁¹⁾ S^(l_k^k). We will write dim(P)ˆ(l₁;. . .;l_k). Clearly, dim depends on the ordering of the re- presentatives of the simpleR-modules. Therefore we will always suppose that with any semilocal ringRwe have some fixed ordering on the set of representatives of the simple R-modules. By [13, Theorem 1.3], two pro- jectiveR-modulesPandQare isomorphic if and only if dim(P)ˆdim(Q).

LEMMA 3.1. Let R be a right noetherian semilocal ring. If I and K are idempotent ideals of R such that I‡J(R)ˆK‡J(R), then IˆK. In particular, R has only finitely many idempotent ideals.

PROOF. SinceR=J(R) has only finitely many (idempotent) ideals, it is enough to show thatI‡J(R)ˆK‡J(R) impliesIˆKwheneverIandK are idempotent ideals ofR.

First suppose that IK are idempotent ideals of R. In particular, KIˆI. Suppose that I‡J(R)ˆK‡J(R). Then KˆK(K‡J(R))

ˆK(I‡J(R))ˆI‡KJ(R). Since R is right noetherian, Nakayama's Lemma givesIˆK.

In general, suppose that I and K are idempotent ideals of R with I‡J(R)ˆK‡J(R). Then IandI‡K are idempotent ideals of Rsuch that I‡J(R)ˆI‡K‡J(R). By the previous step, IˆI‡K, and thereforeKI. The proof forIK is similar. p COROLLARY3.2. LetR be a right noetherian semilocal ring. ThenR satisfies condition(*).

PROOF. Let p:R!R=J(R) be the natural projection. Consider a descending sequence of ideals in R such that I_k‡1I_k ˆI_k‡1. Since

p(I₁);p(I₂);. . .is a descending sequence in an artinian ringR=J(R), there

exists k₀2N such that p(I_k)ˆp(I_k0) for every kk₀. Then I_k‡1

ˆIk‡1(Ik‡1‡J(R))ˆI²_k‡1‡Ik‡1J(R) for every kk0. By Nakayama's Lemma, we see that I_k is idempotent for any k>k0. Now conclude by Lemma 3.1.

The following lemma and its application was suggested by Dolors Herbera.

LEMMA 3.3. Let P be a projective R-module with trace ideal I and let S be a simple R-module. The following conditions are equivalent.

(i) S is a factor of IR. (ii) S is a factor of P.

(iii) SIˆS.

PROOF. (i))(ii) Suppose that f:I!S is nonzero. Then f(i)6ˆ0 for some i2I. Since I is the trace ideal of P, there are homomorphisms g1;. . .;g_k:P!I and p1;. . .;p_k2P with g1(p1)‡ ‡g_k(p_k)ˆi. There- forefg_j6ˆ0 for some 1jk. (Observe that we did not usePprojective for this implication.)

(ii))(iii) Follows fromPIˆP.

(iii))(i) Letf:R_R!Sbe nonzero. Thenf(I)ˆS, becauseSIˆS. p PROPOSITION3.4. LetRbe a semilocal left and right noetherian ring.

Suppose that P is a countably generated projective module. Then there exists a least idealIinRsuch thatP=PIis finitely generated.

Moreover, let fS₁;. . .;S_kg be a set of representatives of the simple modules, indexed in such a way that P=PJ(R)'Sⁿ₁¹ Sⁿ_l^l S^(v)_l‡1 S^(v)_k , n1;. . .;nl2N0,0lk. Then:

(i) P is I-big,

(ii) S_iˆS_iI if and only if i>l, (iii) P=PI=rad(P=PI)'Sⁿ₁¹ Sⁿ_l^l.

PROOF. We have seen in Corollary 3.2 that R satisfies (*). By Lemma 2.4, there exists Isuch thatP=PIis finitely generated and Iis contained in any other ideal K such that P=PK is finitely generated.

Moreover, P is I-big according to Corollary 2.10. Since I is finitely generated as a left ideal, there exists a uniqueI-big projective moduleB with trace idealIandPB^(v)'Paccording to Remark 2.1. By Lemma 3.3, if S is a simple module, then S^(v) is a factor of P (and hence of P=PJ(R)) wheneverSIˆS. Choose an enumeration of the simple mod- ules such that S₁;. . .;S_lare annihilated byIandS_l‡1;. . .;S_k are factors of I. Let 0l₁;. . .;l_k 1 be such that P=PJ(R)'S^(l₁¹⁾ S^(l_k^k). As remarked above, l_l‡1ˆ ˆl_kˆ 1. On the other hand, S^(l₁¹⁾ S^(l_l ^l)is a factor ofPannihilated byI, hence a factor ofP=PI.

Thus l1;. . .;ll are finite. Suppose P=PI=rad(P=PI)'Sⁿ₁¹ Sⁿ_l^l. Since S^l₁¹ S^l_l^l is a semisimple factor of P=PI, lini for any 1il. On the other hand, Sⁿ₁¹ Sⁿ_l^l is a factor of P, so that

n_il_i for every 1il. p

Recall that a nonzero module is calledsuperdecomposableif it has no indecomposable direct summand. The following lemma explains our crav- ing for the existence of superdecomposable projectives over semilocal rings.

LEMMA 3.5. Suppose that there exists a superdecomposable projective module over a semilocal ring R. Then R possesses a nonzero decomposable projective module having all its nonzero direct summands isomorphic.

PROOF. By the theorem of Kaplansky, if there exists a super- decomposable projective module, then there exists a superdecomposable countably generated projective module. It follows easily that then there exists a superdecomposable countably generated projective moduleQsuch that dim(Q)ˆ(m₁;. . .;m_k), wherem_iˆ0 orm_iˆvfor any 1ik(use the additivity of dim). LetQ⁰ be a superdecomposable module such that dim(Q⁰) has all components inf0;vg and the number of nonzero compo- nents is as small as possible. Then it is easy to see that dim(Q⁰)ˆdim(Q⁰⁰) for any nonzero direct summand ofQ⁰, so [13, Theorem 1.3] gives thatQ⁰has

the required property. p

The following example discovered by Puninski [12] shows that a su- perdecomposable projective module may exist even over a semilocal noe- therian ring.

EXAMPLE 3.6 (cf. [12, Proposition 7.5]). Let SˆZn2Z[3Z[5Z and letZ_Sbe the localization of integers atS. LetA₅be the group of even permutations on the set of cardinality 5. Then the group ringZS[A5] is a semilocal left and right noetherian ring with a superdecomposable projec- tive module.

PROOF. We will repeat general arguments of [12] that show that the ring RˆZ_S[A₅] is a semilocal left and right noetherian ring. First,Ris a finitely generated as a (left and right) module over the commutative noetherian ring ZS, thereforeRis noetherian on both sides. Further,R'EndR(RR), so that there exists an injective homomorphismW:R!End_ZS(R) given by the left

multiplication ofRonRZS. For anyg2A5, letug2EndZS(R) be given by ug(r)ˆrg;r2R. Obviously, ImW consists exactly of the elements of EndZS(R) that commutewith alltheendomorphismsofthesetfugjg2A5g.

It follows thatWis a local homomorphism, that is,ris invertible inRifW(r) is invertible in End_ZS(R). Finally, since End_ZS(R)'M₆₀(Z_S) is a semilocal ring, the ringRis also semilocal by [4, Theorem 1].

Let Ibe the augmentation ideal of R, that is, the kernel of the epi- morphismf:R!Z_S,f P

g2A5

r_gg

ˆ P

g2A5

r_g. Since [A₅;A₅]ˆA₅, the idealI is idempotent [1, Theorem 2.1]. By [12], it can be proved that every non- zero finitely generated projective module overRis a generator. In fact, we only need to show that if P is a finitely generated projective R-module, then Tr(P) cannot be contained inI: SinceZS is a Dedekind ring of zero characteristic and 2,3,5 are not invertible inZS,P⁰ˆP_ZS[A5]Q[A5] is a free Q[A₅]-module by [16, Theorem 8.1]. If Tr(P)I, then P⁰I⁰ˆP⁰, where I⁰is the augmentation ideal ofQ[A₅], a contradiction. Let Qbe a projective module having trace idealI. IfQ⁰is a nonzero direct summand of Q, thenQ⁰cannot be finitely generated, and there is a nonzero idempotent idealKsuch thatQ⁰isK-big. ThereforeQ⁰cannot be indecomposable. p REMARK3.7. In the next section we look closer at the localizations of Z[A₅] showing that the augmentation ideal of Z_S[A₅] contains no non- trivial idempotent ideals.

4. Integral group rings,especiallyZ[A5].

In this section, we prove that an integral group ring of a finite group satisfies condition (*). The proof presented here is not the quickest one, but it shows how to calculate idempotent ideals in particular examples. We apply this method toZ[A₅] describing all countably but not finitely gen- erated projective modules up to isomorphism. Our approach will be ele- mentary.

First of all, let us introduce the notation we will use throughout this section. Let Gbe a finite group andRˆZ[G];R_pˆZ_(p)[G];R₀ˆQ[G].

For any primepwe haveRR_pR₀. IfIis an ideal ofR,I_(p)stands for the ideal inRpgenerated byIandI(0)stands for the ideal ofR0generated byI. That is,I_(p) ˆZ_(p)IandI₍₀₎ˆQI. We say that an idealIR(or an idealIRp)extends toan idealKR0ifK ˆQI. IfSis a commutative ring, the augmentation ideal of S[G] is the kernel of the canonical

homomorphismf:S[G]!Sgiven byf P

g2Gsgg

ˆ P

g2Gsg. It is denoted by Aug(S[G]).

In the following we summarize the framework of our calculations.

FACT 4.1. Let G be a finite group and let RˆZ[G]. Then (i) If I is an ideal of R, then I₍₀₎ˆQI_(p) for every prime p.

(ii) Let I;K be ideals in R. Then IˆK if and only if I(p) ˆK(p) for every prime p.

(iii) If IR is an ideal, then I is idempotent if and only if I_(p) is idempotent for every prime p.

(iv) If I;K are idempotent ideals of R and p a prime not dividingjGj, then I(p)ˆK(p) if and only if I(0)ˆK(0). In this case, all central idempotents of R0are contained in Rpand every idempotent ideal of Rpis generated by a central idempotent.

(v) Let e be a central idempotent of R₀ and suppose that, for every prime p that dividesjGj, there is an idempotent ideal I_pR_pwith QIpˆeR0. Then there exists a unique idempotent ideal IR such that I(p) ˆIpfor any pj jGjand I(p)ˆeRpfor any p6 j jGj.

PROOF. Statements (i), (ii), (iii) and (v) are rather standard. Statement (iv) follows from the fact thatZ_(p)[G] is a maximalZ_(p)-order inQ[G] if and only ifpdoes not dividejGj(see [5, Proposition 27.1]) and using the ma- chinery of maximal orders.

Here we give another proof of (iv). Let QF be a finite Galois ex- tension ofQsuch thatFis a splitting field ofG. Recall that ifjis a complex character of a simple representation ofGoverF(considered as a function j:G!F), then j(1_G)

jGj P

g2Gj(g ¹)g

is a primitive central idempotent of F[G]. In order to get the set of primitive central idempotent of Q[G], consider the usual action of Gal(F:Q) on the set of primitive central idempotents ofF[G] and take sums of the orbits. It follows that if pis a prime andp6 j jGj, then any central idempotent ofR0is inRp.

LetIbe an idempotent ideal ofRp, wherepis a prime not dividingjGj.

ThenQIis an ideal ofR₀generated by a central idempotenteofR₀. Since e2R_p,K ˆeR_p is an idempotent ideal ofR_p, necessarilyIK because eIˆI. SinceQIˆQK, there existsk2Nsuch thatp^kKI. AsZ_p[G] is semisimple, idempotent ideals inZpⁿ[G] are generated by central idempo- tents for anyn2N(combine [2, Proposition 27.1] and [10, 22.10]). Moreover, it is easily seen that ifK⁰is an idempotent ideal ofZ_p²ⁿ[G], thenpⁿK⁰is an

essential submodule of K⁰. Now letp:Rp!Z_p^2k[G] be the canonical pro- jection. Thenp^kp(K)p(I)p(K). By our previous remarks,p(I)ˆp(K).

Since Rp is a semilocal noetherian ring and p is an epimorphism with Ker pJ(R_p) (Fact 4.3),IˆKfollows from Lemma 3.1. p

The following result also follows from [15, Theorem 3].

COROLLARY4.2. Any integral group ring of a finite group satisfies(*) and has only finitely many idempotent ideals.

PROOF. SinceRis a ring of Krull dimension 1, it is enough to see that Rhas no descending chain of idempotent ideals. LetIbe an idempotent ideal, let e be a central idempotent of R₀ such that eR₀ˆQI. Then I_(p)ˆeR_pfor every primepnot dividingjGjby Fact 4.1(iv). Ifpis a prime divisor of jGj, then we have only finitely many possibilities for I_(p) by Lemma 3.1. Therefore, by Fact 4.1(v), R contains only finitely many

idempotent ideals. p

The proof of Corollary 4.2 shows a method of finding idempotent ideals inR. We can proceed as follows: Take an idealI₀ofR₀. LetPbe the set of prime divisors ofjGj. For anyp2P, determine the setMpconsisting of the idempotent ideals of Rp that extend to I0. Then there is a bijective cor- respondence between the idempotent ideals ofRextending toI₀and the set Q

p2PM_p.

Thus we can now work in semilocal localizations (see [5] or use the same kind of arguments as in Example 3.6).

FACT 4.3. The natural homomorphism p_p:R_p!Z_p[G] is a local morphism for any prime p. In particular, pRpJ(Rp) and Rp is a semilocal ring.

Let us show the method in the case ofGˆA5, the alternating group on 5 elements. The usual question ``WhyA5?'' has a simple answer. By a result of Swan [17], non-finitely generated projective modules over integral group rings of finite solvable groups are free. Therefore there are no proper idempotent ideals in integral group rings of finite solvable groups (a direct proof of this was given by Roggenkamp [14]). On the other hand, it is known [1] that if G contains a perfect normal subgroup H, that is, [H;H]ˆHandH/G, then the augmentation ideal ofH(that is, the kernel

of the canonical homomorphismZ[G]!Z[G=H]) is idempotent. If there were no other idempotent ideals inZ[G], then countably generated pro- jective modules over Z[G] would be induced by finitely generated pro- jective modules overZ[G=H], whereHranges in the set of perfect normal subgroups ofG. So A₅ is the first candidate to check. Unfortunately, we will see that there indeed exists an idempotent ideal that is not the aug- mentation ideal of a perfect normal subgroup. Hence the structure theory for big projective modules over integral group rings seems to be more complicated.

Throughout the next paragraphs, suppose GˆA₅. The conjugacy classes ofGare the following:c₁- the conjugacy class of the identity;c₂- the permutations that are product of two 2-cycles (the conjugacy class of (1;2)(3;4));c3- all 3-cycles;c5- the conjugacy class of (1;2;3;4;5); andc⁰₅- the conjugacy class of (1;3;5;2;4).

Let us recall what we know about the semisimple ringR₀. The primitive central idempotents ofR₀aree₁ˆ 1

60 P

g2Gg,e₃ˆ 1 20

6 2P

g2c2

g‡ P

g2c5[c⁰₅g , e₂ˆ 1

15

4‡ P

g2c3

g P

g2c5[c⁰₅g

,e₅ˆ 1 12

5‡P

g2c2

g P

g2c3

g

. LetT₁;T₃;T₂;T₅ be the corresponding simple modules (e_icorresponds toT_i). Their dimensions overQare 1;6;4;5.

We need to calculate the idempotent ideals in R₂;R₃;R₅. Set S_iˆZ_i[A₅] foriˆ2;3;5. By Fact 4.3, any simple S_i-module can be con- sidered as a simpleRi-module and there are no other simpleRi-modules except for these. In order to find the number of different simple modules overR_i, one can use the following results proved by Berman and Witt (see [5, Theorem 21.5, Theorem 21.25]).

FACT 4.4. Let G be a finite group of exponent m.

(i) Letbe the relation on G given by gh if g is conjugate to h^tfor some t2N;(t;m)ˆ1. Then the number of simpleQ[G]-modules is equal tojG= j.

(ii) Let p be a prime, and Gp⁰the set of p-regular elements of G. On the set Gp⁰consider the equivalencedefined by gh if g is conjugate to h^pj for some j2N₀. Then the number of simple Z_p[G]-modules is equal to jG_p⁰= j.

Thus each ring R₂;R₃;R₅ has exactly three non-isomorphic simple modules. Now idempotent ideals in semilocal rings are determined by their

simple factors (Lemma 3.1). Call a ring T almost semiperfect if for any simple T-module Mthere exists a positive integernsuch that Mⁿ has a projective cover. The next lemma describes the distribution of idempotent ideals inR_i, fori2 f2;3;5g. In all the remaining proofs of this section,I_i stands for Aug(R_i).

LEMMA 4.5. Let i2 f2;3;5g. The ring R_i has exactly 3 minimal idempotent ideals and any nonzero idempotent ideal of Ri is a sum of minimal idempotent ideals. Moreover, Riis almost semiperfect and any idempotent ideal of R_i is a trace ideal of a finitely generated projective module. Finally, two minimal idempotent ideals are described as follows:

If I_iis the augmentation ideal of R_i, then e_iR_iand(1 e_i)I_iare minimal idempotent ideals of Ri.

PROOF. We give the proof for iˆ5, the remaining cases are similar.

The augmentation idealI5R5is idempotent, becauseA5is perfect. Ob- servee52R5. Therefore alsoe5R5and (1 e5)I5are idempotent ideals. Let M₁;M₂;M₃ be the set of representatives of the simple R₅-modules and suppose thatM₁is the module induced by the trivial representation ofS₅. Obviously,M₁I₅ˆ0, soM₁is not a factor ofI₅. SinceI₅must have at least two simple factors (it contains two different nontrivial idempotent ideals), M2;M3 are both factors ofI5. Choose the notation in such a way thatM2

is the unique simple factor of (1 e5)I5andM₃is the unique simple factor of e₅R₅.

Obviously, e₅R₅ is the trace ideal of the projective module e₅R₅. Set gˆ(1;2)(3;4). The idempotente⁰ˆ(1 e₅)

1 1

2(1‡g)

gives a projec- tive R₅-module P⁰ˆe⁰R₅ with trace ideal (1 e₅)I₅. It follows that P⁰=P⁰J(R₅)ˆM^k₂, for somek2N(it is necessary to check thatP⁰6ˆ0, below we calculateZ(5)-rank ofP⁰using the so called Hattori-Stallings map).

On the other hand, the projective modulePˆ(1 e5)R5has the radical factorP=PJ(R5)ˆM1M^l₂. ThereforeP^0lsplits inP^k, that is, there exists a projective module Q such thatP^kˆP^0lQ. Since Q=QJ(R₅)'M₁^k, it follows that Tr(Q) is an idempotent ideal such thatM₁ is its only simple factor.

So we have proved that the finitely generated projective modules Q;P⁰;e5R5 are the projective covers of convenient finite powers of M₁;M₂;M₃ and R₅ is almost semiperfect. Therefore Tr(Q), Tr(P⁰) and Tr(e₅R₅) are the minimal idempotent ideals ofR₅ and any nonzero idem- potent ideal ofR₅ is a sum of minimal idempotent ideals. p

LEMMA 4.6. The only idempotent ideals of RˆZ[A5] contained in Aug(R)are0andAug(R).

PROOF. Set IˆAug(R) and let 06ˆK be an idempotent ideal of R contained inI. ThenK_(i)also is a non-zero idempotent ideal ofR_icontained in Ii, hence, by Lemma 4.5, QK(i) is either eiR0, (e2‡e3‡e5 ei)R0 or I(0)ˆ(e2‡e3‡e5)R0. Now QK(2)ˆQK(3)ˆQK(5)ˆQK. An easy in- spection gives that the only possibility is K_(i)ˆI_i for any i2 f2;3;5g.

ThereforeKˆIby Fact 4.1(v). p

For anyi2 f2;3;5g, letK_ibe the (unique) minimal idempotent ideal of Rithat is not contained in the augmentation ideal ofRi. In order to classify the idempotent ideals inRthat are not contained in the augmentation ideal of R, we must determine QK₂;QK₃ andQK₅. Let us prove an auxiliary general result, which is probably well known.

LEMMA 4.7. LetW:S!T be a ring homomorphism. If P is a projective S-module with trace ideal I, then P_ST is a projective T-module with trace ideal TW(I)T.

PROOF. Let X be a set and let p:S^(X)!S^(X) be an idempotent en- domorphism ofS^(X) such that p(S^(X))'P. If pis expressed as a column- finite idempotent matrixA(with respect to the canonical basis), thenW(A) is an idempotent matrix corresponding to the endomorphismp⁰:T^(X)!T^(X) such that PST'p⁰(T^(X)). Now Tr(P) (resp. Tr(PST)) is the ideal

generated by the entries ofA(resp.W(A)). p

FACT 4.8. Let S be a commutative local ring and let H be a finite group. Suppose that eˆ P

h2Hshh is an idempotent of S[H]. The module eS[H]is free when considered as an S-module. Moreover,jHjs1ˆn1_S, where n2N₀is the rank of the free S-module eS[H].

PROOF. This is a consequence of [7, Example 7]. Let us briefly explain the idea. LetTbe a ring andT=[T;T] be the group that is the factor of the additive group ofT modulo [T;T]ˆ hft₁t₂ t₂t₁jt₁;t₂2Tgi_(T;‡). There exists a map r:K0(T)!T=[T;T] defined as follows. Let P be a finitely generated projective module over T and A an idempotent matrix re- presentingP. Thenr([P]):ˆTr(A)‡[T;T] (here Tr(A) is the sum of the diagonal entries inA).