New York Journal of Mathematcs New York J. Math. 11 (2005) 21 33. Characterzaton of matrx types of ultramatrcal algebras Gábor Braun Abstract. For any equvalence relaton on postve ntegers such that nk mk f and only f n m, there s an abelan group G such that the endomorphsm rng of G n and G m are somorphc f and only f n m. However, G n and G m are not somorphc f n m. Contents 1. Introducton 21 1.1. Dmenson groups 22 1.2. Ultramatrcal algebras 22 2. Equvalence of dmenson groups and ultramatrcal algebras 23 3. Overvew of the constructon 24 4. Notaton 25 5. Abelan groups wth prescrbed endomorphsms 25 6. Constructon of the dmenson group 26 6.1. The abelan group under the dmenson group 26 6.2. The partal order 28 7. Automorphsms of dmenson groups 31 References 33 1. Introducton We construct partally ordered abelan groups such that the orbt of a dstngushed element under the automorphsm group s prescrbed; the precse statement s Theorem 1.1 n Subsecton 1.1. The prescrbed orbt controls the matrx type of a rng,.e., whch matrx algebras over the rng are somorphc, hence we can characterze the matrx types of ultramatrcal algebras over any prncpal deal doman, see Theorem 1.2 n Subsecton 1.2. If the ground rng s Z then these algebras Receved October 6, 2004; revsed January 5, 2005. Mathematcs Subject Classfcaton. Prmary 20K30, 16S50; Secondary 06F20, 19A49. Key words and phrases. matrx type of a rng; dmenson group; ultramtarcal algebra; automorphsm group of a dmenson group. 21 ISSN 1076-9803/05
22 Gábor Braun are realzable as endomorphsm rngs of torson-free abelan groups, whch groups therefore have the property stated n the abstract, see Corollary 1.3. We are ndebted to Péter Vámos who draw our attenton to ths wonderful problem. 1.1. Dmenson groups. An order unt n a partally ordered abelan group s a postve element u such that for every element x there s a postve nteger n such that nu x. A dmenson group (D,,u) s a countable partally ordered abelan group (D, ) wth order unt u such that every fnte subset of D s contaned n a subgroup, whch s somorphc to a drect product of fntely many copes of (Z, ) as a partally ordered abelan group. Our man result, whch wll be proven from Secton 3 on, s: Theorem 1.1. Let H Q + be a subgroup of the multplcatve group of the postve ratonal numbers. Then there s a dmenson group (D,,u) whose group of order-preservng automorphsms s somorphc to H. Furthermore, under ths somorphsm every element of H acts on u by multplcaton by tself as a ratonal number. In the specal case when H s generated by a set S of prme numbers, one may choose D to be the rng Z[S 1 ] and u = 1, see [7, Proposton 4.2]. 1.2. Ultramatrcal algebras. An ultramatrcal algebra over a feld or prncpal deal doman F s an F -algebra whch s a unon of an upward drected countable set of F -subalgebras, whch are drect products of fntely many matrx algebras over F. 1.2.1. Matrx types of rngs. Let M n (R) denote the rng of n n matrces over the rng R. Obvously, M n (R) s ultramatrcal f R s ultramatrcal. The matrx type of a rng R s the equvalence on postve ntegers descrbng whch matrx algebras over R are somorphc: (1) mt(r) :={(n, m) M n (R) = M m (R)}. Clearly, f (n, m) mt(r) then (mk, nk) mt(r) for all postve ntegers m, n, k. The converse also holds for ultramatrcal algebras but probably not for all rngs. The next theorem states that all such equvalences ndeed arse as matrx types of ultramatrcal algebras: Theorem 1.2. Let F be a feld or prncpal deal doman and be an equvalence relaton on the set of postve ntegers. Then the followng are equvalent: () For all postve ntegers n, m and k (2) n m nk mk. () There s a (unque) subgroup H of the multplcatve group Q + ratonal numbers such that for all postve ntegers n and m: of postve (3) n m n m H. () There exsts an ultramatrcal F -algebra wth matrx type.
Matrx types of ultramatrcal algebras 23 The second statement s clearly a reformulaton of the frst one, whch s useful for explct constructon of equvalences, as ponted out by the referee. The equvalence of the second and thrd statements s a smple consequence of our man result, Theorem 1.1, as we wll explan n the next secton. 1.2.2. Bass types of rngs. In a smlar ven, the bass type of a rng R characterzes whch fnte rank free modules are somorphc: (4) bt(r) :={(n, m) R n = R m }. The analogue of Theorem 1.2 for bass types s Theorem 1 of [3]: every equvalence relaton wth the property n m f and only f n + k m + k s the bass type of a rng of nfnte matrces. Clearly, f R n = R m then R n+k = R m+k. All equvalence relaton wth ths property arse as bass type of a rng: see [6]. The bass type s obvously smaller than the matrx type. It seems plausble that ths s the only relaton between the two types. For example, ultramatrcal algebras have trval bass type.e., free modules of dfferent fnte rank are not somorphc. Every ultramatrcal algebra R over F = Z s a countable reduced torson-free rng, and hence s the endomorphsm rng of a torson-free abelan group G by [1, Theorem A]. Then M n (R) s the endomorphsm rng of G n. Obvously G n and G m are not somorphc f n m snce there s no nvertble n m matrx over R. Hence we have the statement n the abstract as an mmedate corollary to the last theorem: Corollary 1.3. Let be an equvalence relaton on postve ntegers wth the property that n m f and only f nk mk for all postve ntegers n, m and k. Then there s a torson-free abelan group G such that the endomorphsm rng of G n and G m are somorphc f and only f n m. Moreover, G n and G m are somorphc f and only f n = m. 2. Equvalence of dmenson groups and ultramatrcal algebras Dmenson groups and ultramatrcal algebras over a fxed feld or prncpal deal doman are essentally the same. In ths secton, we recall ths equvalence, whch shows that Theorem 1.2 follows from Theorem 1.1. For more detals and proofs see [7, Proposton 4.1] or [5, Chapter 15, Lemma 15.23, Theorems 15.24 and 15.25], whch assume that the ground rng s a feld, but the arguments also work when t s a prncpal deal doman. Frst we defne the functor K 0 from the category of rngs to the category of preordered abelan groups wth a dstngushed order unt. The somorphsm classes of fntely generated projectve left modules over a rng R form a commutatve monod where the bnary operaton s the drect sum. The quotent group of the monod s denoted by K 0 (R). Declarng the somorphsm classes of projectve modules to be nonnegatve, K 0 (R) becomes a preordered group. The somorphsm class u of R, the free module of rank 1, s an order unt of K 0 (R).
24 Gábor Braun If f : R S s a rng homomorphsm then let K 0 (f) map the somorphsm class of a projectve module P to that of S R P. So K 0 (f) s a homomorphsm preservng both the order and the order unt. If R s an ultramatrcal algebra then K 0 (R) s a dmenson group. Conversely, every dmenson group s somorphc to the K 0 of an ultramatrcal algebra. If R and S are ultramatrcal algebras then every morphsm between K 0 (R) and K 0 (S) s of the form K 0 (f) for some algebra homomorphsm f : R S. However, f s not unque n general. Nevertheless, every somorphsm between K 0 (R) and K 0 (S) comes from an somorphsm between R and S. Thus, by restrcton, K 0 s essentally an equvalence between the category of ultramatrcal algebras over a gven feld and the category of dmenson groups wth morphsms the group homomorphsms preservng both the order and the order unt. Now we examne how the matrx type of an ultramatrcal algebra can be recovered from ts dmenson group. The standard Morta equvalence between R and M n (R) nduces an somorphsm between the K 0 groups. However, ths somorphsm does not preserve the order unt, n fact K 0 (M n (R)) s (K 0 (R),,nu) where u s the order unt of K 0 (R). So f R s an ultramatrcal algebra, then the dmenson groups K 0 (M n (R)) and K 0 (M m (R)) (and hence the algebras M n (R) and M m (R)) are somorphc f and only f there s an order-preservng automorphsm of K 0 (R) sendng nu to mu. Obvously, for any dmenson group (D,,u) whether an order-preservng automorphsm maps nu to mu, depends only on the factor m/n. Such factors m/n form a subgroup of the multplcatve group Q + of the postve ratonals. So the classfcaton of matrx types of ultramatrcal algebras s equvalent to the classfcaton of subgroups of Q + whch arse from dmenson groups n the above constructon. Theorem 1.1 states that all subgroups arse, and Theorem 1.2 s just the translaton of t to the language of matrx types of ultramatrcal algebras. 3. Overvew of the constructon In the rest of the paper we prove Theorem 1.1. In ths secton we outlne the man deas of the proof and leave the detals for the followng sectons. The next secton fxes notatons used frequently n the rest of the paper. Secton 5 recalls a constructon of abelan groups. The actual proof s contaned n the rest of the sectons, whch are organzed so that they can be read ndependently. At the begnnng of every secton, we shall refer to ts man proposton, whch wll be the only statement used n other sectons. The same s true for subsectons. To prove Theorem 1.1, we fx a subgroup H of the postve ratonals and construct a dmenson group D for t. We search for D (as an abelan group wthout any order) n the form D := Qu G where H acts on the drect sum componentwse. We let H act on Qu by multplcaton as requred to act on the order unt. A key observaton (Proposton 7.1) s that f the only automorphsms of G are the elements of H and ther negatves, then for any partal order on D, whch s preserved by H and makes (D,,u) a dmenson group, the order-preservng automorphsm group of D s only H. So(D,,u) satsfes the theorem. Therefore, all we have to do s to fnd such a G and. Actually, G s already constructed by A. L. S. Corner n [1]. Snce we shall use the structure of G to defne
Matrx types of ultramatrcal algebras 25 the partal order, we recall a specal case of the constructon n Secton 5, whch s enough for our purposes. See also [4] for the general statement. Fnally, we defne a partal order on D makng t a dmenson group n Subsecton 6.2. The basc dea s to explctly make some subgroups of D order-somorphc to Z n and show that these partal orders on the subgroups are compatble. Ths wll be based on a descrpton of G (Proposton 6.1): we defne elements of G, whch wll be bases of order-subgroups Z n of D and state relatons between them mplyng the compatblty of partal orders. 4. Notaton Let x h denote the element of the group rng ZH correspondng to h H. Ths s to dstngush x h a the value of x h at a from ha the element a multpled by the ratonal number h. If the automorphsm group of an abelan group s the drect product of a group H and the two element group generated by 1 then we say that the automorphsm group s ±H. 5. Abelan groups wth prescrbed endomorphsms We revse a specal case of A. L. S. Corner s constructon of abelan groups wth prescrbed endomorphsm rngs. I am grateful to Rüdger Göbel and hs group for teachng me ths method. Let ˆM denote the Z-adc completon of the abelan group M. The followng result s s a specal case of Theorem 1.1 from [2] by takng A = R and N k =0: Proposton 5.1. Let R be a rng wth free addtve group. Let B be a free R- module of rank at least 2 and {w b : b B \{0}} a collecton of elements of Ẑ algebracally ndependent over Z. Let the R-module G be (5) G := B,Rbw b : b B \{0} ˆB. Then G s a reduced abelan group wth endomorphsm rng R. Here means purfcaton:.e., we add all the elements x of ˆB to E := B b B\{0} Rbw b for whch nx E for some nonzero nteger n. The usual down-to-earth descrpton of G s the followng, whch we shall use n Subsecton 6.1: we select postve ntegers m n such that every nteger dvdes m 1...m n for n large enough. We choose elements w (n) b of Ẑ for all nonzero elements b of B and natural numbers n such that w (0) b = w b and w (n) b m n w (n+1) b s an nteger. Then G s generated by the submodules B and Rbw (n) b. Obvously, G n := B b B\{0} Rbw (n) b s a submodule of G and these modules G n form an ncreasng chan whose unon s the whole G. Remark 5.2. Note that the automorphsm group of G s the group of unts of R, whch s just ±H f R = ZH and H s an orderable group. (It s a famous conjecture that the group of unts of ZH s ±H for all torson-free groups H.) We wll be nterested n the case when H Q + and R = ZH s a group rng. The free module B wll have countable rank. Recall that there are contnuum many elements of Ẑ algebracally ndependent over Z, so the constructon works n ths case, and we wll get a reduced abelan group G wth automorphsm group ±H.
26 Gábor Braun 6. Constructon of the dmenson group We now construct our dmenson group D = Qu G startng from a subgroup H Q + of the postve ratonals. The man result of the secton s Proposton 6.2. The proposton n Subsecton 6.1 defnes G and summarzes the (techncal) propertes of G used n Subsecton 6.2 to put the partal order on D. Let n 2 denote the set of sequences of length n whose elements are 0 or 1. If y s a fnte sequence of 0 and 1 then let y0 denote the sequence obtaned from y by addng an addtonal element 0 to the end. Smlarly, we can defne y1. These sequences wll be dentfed wth elements of G. 6.1. The abelan group under the dmenson group. In ths subsecton we construct an abelan group G satsfyng the followng propertes. Essentally, G wll be the underlyng abelan group of our dmenson group. Proposton 6.1. Let H Q + be a subgroup of the multplcatve group of the postve ratonal numbers. Let R := ZH denote ts group rng. Then there exst: an R-module G, whch s also a reduced abelan group, a fnte subset F n H for all postve nteger n, postve ntegers s n, t n, k n and l n for n 0, subject to the followng condtons: () Aut G = ±H. () Structure of G: (a) G s a unon of an ncreasng sequence of free submodules G n. (b) G n has base n 2 { c (n) : =1,...,n }. () Relatons descrbng the ncluson G n G n+1 : (a) y = y0+s 2 n y1 for all y n 2. (b) There are ntegers n () h,y for all n, h H and y n 2 such that (6) c (n) s n t n c (n+1) = n () h,y x h y0, h F 0 n () h,y <s nt n. (v) Propertes of s n, t n, k n and l n : (a) k 0 =1, k n+1 = s n k n, l 0 =1, l n+1 = t n l n. (b) Every postve nteger dvdes k n and l n for n large enough. (c) k n (s 2 n n 1) l n s n t n =1 h F h. Proof. The constructon of the tems s easy. One has to take care to defne them n the correct order. Let B be a countable-rank free module over the group rng R := ZH.e., B = ZH A where A s a free abelan group of countably nfnte rank. Usng ths group we defne G by Equaton (5) as n Proposton 5.1. (The Z-adc ntegers w b can be arbtrary.) The proposton tells us that G s a reduced abelan group and Aut G = ±H so () s satsfed. We dentfy a base of A wth the fnte sequences of 0 and 1 not endng wth 0 (so wth the sequences endng wth 1 and the empty sequence). In ths base, the
Matrx types of ultramatrcal algebras 27 sequences of length at most n s a bass of a free subgroup A n of A and the free R-module B n = R A n. Let us enumerate the elements of B \{0} nto a sequence b 1,b 2,... such that b n B n. Every element of B can be wrtten unquely as h H x hb h where b h A and only fntely many of the b h are nonzero. We defne the support of an element of B as the fnte set [ ] (7) x h b h := {h H b h 0} (b h A). h H Let F := [b ] be the support of b. Now we are ready to defne our postve ntegers s n, t n, k n and l n. Let us mpose the followng addtonal condton on them: ( ) t n s dvsble by n, and t n dvdes s n. Now the ntegers can be defned recursvely such that (v)(a), (v)(c) and ( ) hold. These automatcally mply the truth of (v)(b). Now we dentfy the sequences of 0 and 1 wth elements of A. We have already done ths for the sequences not endng wth zero: they form a bass of A. As dctated by ()(a), we set (8) y0 :=y s 2 n y1 y n 2. (Ths s n fact a recursve defnton on the length of y snce y may also end wth zero.) Thus the sequences of fnte length are dentfed wth elements of A such that ()(a) holds and the sequences of length n form a bass of A n as can be easly seen by nducton on n. We turn to the defnton of the G n. Let us choose Z-adc ntegers w (n) for 1 n such that (9) w (0) := w b, w (n) s n t n w (n+1) Z, w (n) Ẑ. We let G n be the free submodule (10) G n := B n n =1 Rb w (n). It follows from the defnton of G (Equaton (5)) that the groups G n form an ncreasng sequence of submodules whose unon s G. The only mssng enttes are the elements c (n). We could set c (n) = b w (n) to satsfy ()(b) but ths may not be approprate for ()(b). Therefore we shall set (11) c (n) := b w (n) + b (n) b (n) B n, n for some b (n), whch wll also satsfy ()(b). For fxed, we are gong to defne the recursvely for n subject to: b (n) (A) b (n) B n. (B) [b (n) ] [b ]=F.
28 Gábor Braun (C) For sutable ntegers n () h,y : ( ) (12) b w (n) s n t n w (n+1) + b (n) = s n t n b (n+1) + n () h,y x h y0, h F, 0 n () h,y <s nt n. Note that the last equaton s just a reformulaton of ()(b) n terms of the b (n). Now we carry out the recursve defnton. We can start wth b () := 0. Observe that (C) determnes how to defne b (n+1) : the left-hand sde s an element of B n+1, a free abelan group wth bass x h y for h H and y a sequence of 0 and 1 of length n + 1. We dvde the coeffcent of every x h y by s n t n. The quotent gves the coeffcent of x h y n b (n+1) and the remander s a coeffcent of the bg sum on the rght. Snce the support of the left-hand sde s contaned n F by nducton, the same s true for b (n+1) and the sum on the rght-hand sde. The left-hand sde s actually contaned n B n not only B n+1. Ths means that the coeffcents of sequences of length n + 1 endng wth 1 are dvsble by s 2 n by ()(a) and hence by s n t n snce t n dvdes s n. So n the sum on the rght-hand sde, the coeffcent of sequences endng wth 1 s zero. Thus we have defned b (n+1) accordng to the requrements. 6.2. The partal order. In ths subsecton we defne a partal order on D whch wll make t a dmenson group. Proposton 6.2. Let H Q + be a subgroup of the multplcatve group of the postve ratonal numbers actng on Qu by multplcaton. Suppose G s a group satsfyng the condtons of Proposton 6.1. Let H act on D := Qu G componentwse. Then there s a partal order on D such that (D,,u) s a dmenson group on whch H acts by order-preservng automorphsms. The dmenson group D n the proposton satsfes all requrements of Theorem 1.1. We shall see n the next secton that the group of order-preservng automorphsms of D s exactly H. The other requrements are obvously satsfed. Proof. We defne the partal order on a larger group, the dvsble hull QD of D. For all natural number n and fnte subset F of H we defne a subgroup of D: D n,f := Z u n (13) Zx h y Zx h c (n). k n h F =1 h F We defne the partal order on QD n,f as the product order (QD n,f, ) :=(Qv n,f, ) n (14) (Qx h y, ) (Qx h c (n), ) h F =1 h F where v n,f := u ( (15) h 1 x h k n y + l n k n h F Note that u s an order unt of QD n,f. n =1 c (n) ).
Matrx types of ultramatrcal algebras 29 The subgroups D n,f form a drected system whose unon s the whole D. It follows that the subgroups QD n,f form a drected system whose unon s QD. We are n a poston now to reduce the proof to two lemmas stated below. The frst one states that the nclusons between the QD n,f are order-embeddngs. It follows that QD has a unque partal order whch extends the partal order of all the QD n,f. Under ths partal order u s clearly an order-unt. The partal order s preserved by H snce any h H maps QD n,f bjectvely onto QD n,hf and ths bjecton s an order-somorphsm of the two subgroups. (Note that x h v n,f = hv n,hf.) The second lemma clams that a cofnal set of the D n,f s order-somorphc to a drect product of fntely many copes of (Z, ), and thus (D,,u) have all the propertes clamed. All n all, the proposton s proved modulo the followng two lemmas. Lemma 6.3. The nclusons between the subgroups QD n,f are order-embeddngs. Lemma 6.4. A cofnal set of the groups D n,f s order-somorphc to a fnte drect power of (Z, ). We consder frst the nclusons. Proof of Lemma 6.3. We clam t s enough to prove that QD n,f s an ordersubgroup of QD n+1,f f F contans F and FF for n. Because t wll follow by nducton on m n that for every n and F and m>nthere s a fnte subset S of H such that QD n,f s an order-subgroup of QD m,f f F contans S. Hence, f QD n,f s a subgroup of QD k,c then both are order-subgroups of QD m,f for sutable m and F, hence QD n,f must be an order-subgroup of QD k,c. Now we prove the clam that QD n,f s an order-subgroup of QD n+1,f f F contans F and FF for all n. For ths, t s good to have the followng general example of an order-embeddng of (Q, ) m nto (Q, ) m+k gven by a matrx: > 0 0... 0 0... 0 0 > 0 0 (16). 0... 0................. 0... 0 > 0 0... 0 On the frst m coordnates ths s an order-somorphsm: every coordnate s multpled by a postve number. On the last k coordnates the map s an arbtrary order-preservng map. By permutatng the coordnates, we may complcate the map. All n all, we see that a homomorphsm (Q, ) m (Q, ) M s an order-embeddng f the canoncal bass elements of the doman have only nonnegatve coordnates n the codoman and every bass element has a postve coordnate, whch coordnate s zero for the other bass elements. We show that the ncluson of QD n,f nto QD n+1,f s an order-embeddng of the above type usng the drect product decomposton (14). To ths end, we express the generators of QD n,f as lnear combnaton of the generators of QD n+1,f (whch
30 Gábor Braun n partcular shows that QD n,f s really a subgroup of QD n+1,f ): (17) x h y = x h y0+s 2 nx h y1 y n 2,h F (18) (19) x h c (n) = s n t n x h c (n+1) + n () t,yx ht y0 t F v n,f = s n v n+1,f + h F s n l n+1 h 1 x h c (n+1) n+1 + + h F h F, n t F, k n (s 2 n 1)h 1 x h y0+ l n n () t,yh 1 x ht y0. h F \F n h F \F y n+1 2 n, h F s n l n+1 h 1 x h c (n+1) k n s 2 nh 1 x h y These are easy consequences of the formulas under () of Proposton 6.1 and the defnton (15) of v n,f. All the coordnates of the above generators are obvously nonnegatve except for the coeffcent of x h y0ofv n,f for y n 2 and h F. So let us consder the coeffcent of h 1 x h y0 n Equaton (19): from the second row comes k n (s 2 n 1) or k n s 2 n dependng on whether h s contaned n F. From the last row l n n () t,yt comes for all t F and n for whch ht 1 les n F. All n all, the coeffcent s at least (20) k n (s 2 n 1) t F, n l n n () t,yt k n (s 2 n 1) t F, n l n s n t n t 0 by (v)(c) from Proposton 6.1. Now we check that each of the above generators of QD n,f has a postve coordnate n QD n+1,f whch coordnate s zero for the other generators. Ths coordnate s x h y1 for x h y where h F and y n 2; t s x h c (n+1) n; fnally, t s v n+1,f for v n,f. for x h c (n) where h F and Thus we have proved that the nclusons between the QD n,f are order-embeddngs. Now we return to our second lemma, namely that a cofnal subset of the D n,f are order-somorphc to a fnte power of Z. Proof of Lemma 6.4. If n s a natural number and F s a fnte subset of H such that for all h H the ratonal numbers k n h 1 and l n h 1 are actually ntegers then the coeffcents of the x h y n (15) are ntegers and hence (21) (D n,f, ) :=(Zv n,f, ) h F (Zx h y, ) n =1 h F (Zx h c (n), ). We show that such D n,f form a cofnal system.e., every D n,f s contaned n a D m,f whch has the above property. Ths s easy once we know that D n,f s contaned n D m,f f m n and F contans F and FF for n. Ths last statement follows form the fact that x h c (n) s contaned n D m,f f m n and h and hf are contaned n F. Ths fact can be proved by nducton on m n: the
Matrx types of ultramatrcal algebras 31 case m = n s obvous because h F.Ifm>nthen x h c (n+1) s contaned n D m,f by nducton and x h (c (n) s n t n c (n+1) ) s also contaned n D m,f by Equaton (6) snce hf s contaned n F. 7. Automorphsms of dmenson groups In ths secton we prove that H s the full group of order-preservng automorphsms of D constructed n Proposton 6.2, whch fnshes the proof of our man theorem. Ths s a specal case of the followng proposton, whch we are gong to prove n ths secton. Note that D/Qu = G has the requred automorphsm group. Proposton 7.1. Let (D,,u) be a dmenson group of rank at least 3 on whch a group H acts by order-preservng automorphsms (the order unt need not be preserved). Let us suppose that the maxmal dvsble subgroup of D s Qu. Furthermore, let us assume that (22) Aut(D/Qu) =±H = Z/2Z( 1) H,.e., the automorphsms of D/Qu are those nduced by H and ther negatves. Then Aut(D, ) =H. In other words, all the order-preservng automorphsms of D are those comng from H. We base our proof on the comparson of multples of u wth elements of D. Ths can be descrbed by some ratonal numbers: Defnton 7.2. Let (D,,u) be a dmenson group. Then for every element d of D we denote by r(d) the least ratonal number q such that qu d. Smlarly, let l(d) denote the greatest ratonal number q wth the property qu d. In other words, for all ratonal numbers q: (23) qu d q r(d), (24) qu d q l(d). We collect the man (and mostly obvous) propertes of the functons r and l n the followng lemma: Lemma 7.3. Let (D,,u) be a dmenson group and d and element of t. Then the followng hold: (a) The numbers r(d) and l(d) exst. (b) We have l(d) =r(d) =0f and only f d =0. (c) r( d) = l(d) and l( d) = r(d). (d) For all ratonal numbers s, (25) r(d + su) = r(d)+s, (26) l(d + su) = l(d)+s. (e) If Φ s an order-preservng automorphsm of D and Φ(u) = qu then (27) l(φ(d)) = ql(d), (28) r(φ(d)) = qr(d). (f) If D has rank at least 3, the functon d l(d)+r(d) from D to the addtve group of ratonal numbers s not addtve. (It s addtve f the rank of D s at most 2.)
32 Gábor Braun Proof. To prove the exstence of l(d) and r(d), we may restrct ourselves to an order-subgroup (Z, ) k contanng d and u. Such a subgroup exsts by the defnton of dmenson group. Clearly, u =(n 1,...,n k ) remans an order unt n the subgroup.e., ts coordnates n are postve. For every element (m 1,...,m k )of(z, ) k the functons r and l are clearly well-defned and have the values (29) (30) r(m 1,...,m k ) = max 1 k l(m 1,...,m k ) = mn 1 k m, n m. n These formulas also show that r(d) =l(d) = 0 f and only f d =0. IfD has rank at least 3 then there s an order-subgroup (Z, ) k of D contanng u wth k 3. We can deduce from the above formulas that r + l s not addtve even when restrcted to such a subgroup. For example, for the elements e whose th coordnate s 1 and all the other coordnates 0, we have r(e 1 )=r(e 2 )=r(e 1 + e 2 )=1andl(e 1 )= l(e 2 )=l(e 1 + e 2 ) = 0, and so r(e 1 + e 2 )+l(e 1 + e 2 ) r(e 1 )+l(e 1 )+r(e 2 )+l(e 2 ). The remanng tems (c), (d) and (e) are obvous. Now we start provng the proposton. Frst we splt the order-preservng automorphsm group of D. Lemma 7.4. Wth the hypothess of Proposton 7.1, let Γ: Aut(D, ) Aut(D/Qu) be the canoncal map,.e., Γ(f) s the automorphsm nduced by f on the quotent. Then there s a semdrect product (31) Aut(D, ) =Γ 1 (Z/2Z) H, where Z/2Z s generated by 1. Proof. Note that Qu s nvarant under automorphsms snce t s the largest dvsble subgroup so Γ s well-defned. Note that the composton (32) H Aut(D, ) Γ Aut(D/Qu) =Z/2Z H H s the dentty, whch mples the clamed decomposton as a semdrect product. Here the frst arrow s the ncluson of H gven by the H-acton on D and the last arrow s projecton onto the second coordnate. Now we show that the frst term of the semdrect product s trval, whch fnshes the proof. To ths end, we choose an order-preservng automorphsm Φ Γ 1 (Z/2Z) and show that t s the dentty. By the choce of Φ, there s a number ε = ±1 such that the mage of Φ ε s contaned n Qu. Moreover, snce Qu s nvarant there s a postve ratonal number q such that Φ(u) = qu. Our frst task s to show that q = 1. Therefore we select a nonzero element d n the kernel of Φ ε. Snce Φ ε maps to a 1-rank group Qu and the rank of D s greater than 1, such an element d exsts. Now we use Lemma 7.3 (e). If ε =1, we obtan r(d) =qr(d) and l(d) =ql(d) and thus q = 1 snce r(d) and l(d) are not both zero. If ε = 1 then r(d) = ql(d) and l(d) = qr(d). Agan, snce at least one of r(d) and l(d) s not zero and q s postve, q must be 1.
Matrx types of ultramatrcal algebras 33 So far we know that Φ(u) =u. Let d be an arbtrary element of D. Then there s a ratonal number s dependng on d such that Φ(d) =εd + su. Our next task s to determne s. We apply Lemma 7.3 agan, but ths tme tem (d) of t. If ε = 1 then r(d) = r(d)+s and l(d) =l(d)+s. We conclude that s = 0 for all d. In other words, Φ s the dentty. If ε = 1 then we have r(d) = s l(d) and l(d) = s r(d). Thus s = r(d)+l(d) for all d, whch means that Φ(d) =d (r(d)+l(d))u. Sor + l s an addtve functon contradctng Lemma 7.3 (f). Hence we have proved that Φ = 1 and ths fnshes the proof. References [1] A. L. S. Corner, Every countable reduced torson-free rng s an endomorphsm rng, Proc. London Math. Soc. (3) 13 (1963), 687 710, MR 0153743 (27 #3704), Zbl 0116.02403. [2] A. L. S. Corner, Endomorphsm rngs of torson-free abelan groups, Proc. Int. Conf. Theory Groups, Canberra 1965, 59-69, 1967, Zbl 0178.02303. [3] A. L. S. Corner, Addtve categores and a theorem of W. G. Leavtt, Bull. Amer. Math. Soc. 75 (1969), 78 82. MR 0238903 (39 #263), Zbl 0188.08502. [4] Rüdger Göbel and Jan Trlfaj, Endomorphsm algebras and approxmatons of modules, Walter de Gruyter Verlag, Berln, to appear, 2005. [5] K. R. Goodearl, Von Neumann regular rngs, second ed., Robert E. Kreger Publshng Co. Inc., Malabar, FL, 1991, MR 1150975 (93m:16006), Zbl 0749.16001. [6] W. G. Leavtt, The module type of a rng, Trans. Amer. Math. Soc. 103 (1962), 113 130, MR 0132764 (24 #A2600), Zbl 0112.02701. [7] Percarlo Mers and Peter Vámos, On rngs whose Morta class s represented by matrx rngs, J. Pure Appl. Algebra 126 (1998), no. 1-3, 297 315, MR 1600554 (99f:16008), Zbl 0899.16003. Alfréd Rény Insttute of Mathematcs, Hungaran Academy of Scences, Budapest, Reáltanoda u 13 15, 1053, Hungary braung@reny.hu Ths paper s avalable va http://nyjm.albany.edu:8000/j/2005/11-2.html.