Webb, M. C.

English

Webb M. C.

University of Szeged

Acta Sci. Math., 41 (1979), 253—190 Nilpotent torsion-free rings and triangles of types M. C. WEBB 1. Introduction. This note modifies an idea of VINSONHALER and WICKLESS [7 ] concerning associative rings having torsion-free additive group. In [7] the necessary and sufficient conditions for a group to support only trivial rings given by REE and WISNER [4] are generalised in such a way that certain groups supporting only nil-potent rings are characterised. In fact more precise information can be obtained giving a bound on the nilstufe of a group. The nilstufe, a notion due to SZELE [6], n(X), of a group X is the largest integer n such that there is an associative ring on X with a non-zero product of n elements. If no such largest integer exists then n(X) = o°. Several authors [1], [3], [5], [8], [9] have obtained bounds for the nilstufe of. a group in certain circumstances. The bound obtained here applies in quite general circumstances. In this note the basic tool of [7] is modified and then used to prove results based on two of the main theorems from [7], in one case giving a considerable generalisation. From now on all groups are torsion-free abelian groups and all undefined concepts are standard from FUCHS [2]. In particular the product of a pair of types t l s t 2 is written txt2 not t j+ to as in [7], and T(X) = {t(x): x£X} is the type-set of the group X. 2. The results. The following is a modification of a definition of [7]. D e f i n i t i o n . A collection of types {t;j.: l^j^n — i+1 , l^i^n} is a triangle of base size n if for any three integers i,j,k satisfying 1 sk^n — i—y'+l, l ^ j ^ n — i + l , I S / ^ K , t.-.jtfc, (;+_••) = ta+k),]-This is strongly related to the definition in [7] but, as will become apparent below, it is easy to handle. For example, the following based on Theorem 2.1 of [7] has a quite straightforward proof. Received July 27, 1977, in revised form April 17, 1978. 254 M. C. Webb T h e o r e m 1. For any torsion-free group X and any associative ring (X, o), a non-zero product of n elements in the ring defines a triangle of base size n from the type-set of X. P r o o f . Suppose that for the elements x l 5 . . . , x„ from X the product x ,o . . .ox n is non-zero. Then denoting by (x)* the pure subgroup generated by an element x of X, the following pure rank 1 subgroups of X are defined: for each pair of in-tegers /, j satisfying 1 1, l ^ i ' s / j , Xi.j = (XjO...OXj + ;_!>*. As a consequence, for the integers i,j,k satisfying 1 s f c s / i — i— j+1, 1 =/= s n - i + 1 , 1 g / S B , t (X ( i + k h j ) = t (Xj0. . .0Xj+ i + t_ 1 ) ^ t (x 7 o. . .ox y + i _ 1 ) t (x y + i o. . .ox J + i + t _ 1 ) . However, t^.jOtC^^j+j-)) = t(xjO ...oxj+i^1)t(xJ+io...oXj+i+k-1) implying H X ^ j ^ t i X f J U X w + J . So if tu=t(XtJ for l s / s n - i + l , 1 ^ / S / i the set of types so defined form a triangle of base size n. Recalling the definition of nilstufe there is the following corollary. C o r o l l a r y . For any torsion-free group X, its nilstufe n(X) is bounded by the maximum base size of triangles that can be formed from T(X). Note that in the Corollary there are no restraints on the group X. Other authors [1], [5], [8] have obtained bounds for n(X) but for groups whose type-sets satisfy some form of chain condition. In most cases the Corollary gives a better bound on the nilstufe and an illustrative example is given following Theorem 2. A second important result of [7] that can now be more easily proved is T h e o r e m 2. If for X= © Xt where each X( has rank 1 a triangle of base size n can be formed from the set {t(Ar,): /£ /} without using the same summand twice, then n(X)^n. P r o o f . Suppose that the triangle is { t j ; : — l s / s n } where the titJ are types of distinct summands Xr Then, where l^k^n—j—i+l, l^j^n—i+1, l ^ / s w . The characteristic of an element x in A" is denoted by / (x ) and so the above inequalities imply that elements xt j from the corresponding rank one summands can be found such that x(x(i+khJ) S y.(Xi,j)x(Xk,(i+J))-i Nilpotent torsion-free rings 255 If Y is the direct sum of the summands of X used in defining the types then a ring (Y, *) can be defined using the following products. For integers i, j, k satisfying l ^ k ^ t i - l + l , l ^ j ^ n — i + l , l S i ' ^ / i , 1 ^ M n . _ i*(i X,,j*Xkii 1 Q (i+k),j if I = i+j if not. These products and the distributive law define a ring which can be shown to be associative. Take three subscripts (kr, k2), (/x, l2), (m1, m2); then ki,ki*xh,lt — { and Also, and 0 if l2 jt kx + k2 xki,kt*xh.it ~i r if I - k 4-k x(ki + li),kt 11 F2 — "IT*! fO if m t ^ k x + l i+kx i + h),k2*Xmi,mi = | ̂ (tj + lj + n _ fO it .... . _ Xh,h*Xm1 ,m2 ~ 1 v if m — 1 4-1 l-x(li + mi),l2 11 m2 — ' l T » 2 l(fci /i *2~ -̂ mi. s | v :f m _ b ,1 i L-*(tl + ll + mi).*2 11 2 — K-y-t li+K^. f m2 h + l2 r 0 if l2 ^ fej + k2 • X * i , * 2 * X ( l i + mi ) , / 2 = | : f / _ j l , 1L JC(*i + /i + mi),t2 11 - Kitlt2-So that the products ( x ^ * * , ^ ) * * ^ , ^ , are non-zero if and only if 12=kx + k2 and m2—Ii+12 in which case they both equal x(k1+il+ml),ki" Furthermore, xlti*xiti* •••*xltB=xtt3l is non-zero. To define an associative ring on X merely take the ring direct sum of (Y, *) and the trivial ring on the complement of Y in X and so n(X)^n. Finally an example is given to illustrate that the bounds obtained using Theorem 1 can be lower than those obtained using other available results. Example . Begin by partitioning the set of all primes into two disjoint infinite subsets and P2 where P2— {Pi,Pi,Ps, ...}. Then define the following subgroups of the rationals. A, = A2 = gp\j:peP1{JP2}, B=gp{-±r,-±r:pZP1, Z+, qiP^, and for each integer i s l , « 256 M. C. Webb For each positive integer m let Xm = © - • • © . Then T(jSrm) = {t(^1), t(fl), t (Q) , . . . , t ( C J } contains a chain of length m + 1 , but no chain of greater length. A type tx in T(X) absorbs a type t2 in T(X) if M ^ t j . It is clear that t(XJ contains no absorbing types so that Proposition 1.2 of [5] implies n(Xm)^m + l. However, a triangle of base length two can be formed from T(Xm), namely t21 = =t(f l) , tu=t(Ai) i = l, 2. So, by Theorem 2, n(Xm)s2. The following lemma will show that no larger triangles can be formed. L e m m a . If X is a group such that T(X) contains no absorbing types the apex of a triangle of base size n formed from T(X) has a chain of length n descending from it in T(X). P r o o f . The proof is by induction on n, the result being trivial for n = 1. Suppose n > l . Let t ^ be the apex of a triangle of base size k, then the apex of any triangle of base size (n — 1) from T(X) has a chain of length («—1) descend-ing from it in T(X). If the triangle is {t(- .: — l^i^n} then tn,l — t(n-l),iti in. Now .t(n_1} j is the apex of a triangle of base size (n — 1) from T(X) so is the maximal type of a chain of length (« — 1) in T(X). Suppose t„;i = t(„_1)1 then t ( n_1 ) 1t l f] = - t o i - i M a n d i i s a n absorbing type in T{X). Thus ^ and so has a chain of length n descending from it in T(X). Returning to the example it should be noted that t(B) has only chains of at most length 2 descending from it in T(X„). Furthermore by considering the summands of Xm it is clear that in any ring on Xm, X2QB since PlC\Pi is empty. Since any product of more than two elements in any ring on X is in B, t(B) is the apex of all triangles formed by applying Theorem 1. Hence n (Xm) ^ 2 which with the earlier work gives n(Xm)=2 for all m^l. In conclusion it is noted that this technique fails to work for non-associative rings as the bracketing of a product may be quite arbitrary. Essential to the above is that for a product in an associative ring t((ab)c)=t(a(bc)). Nilpotent torsion-free rings 257 References [1] S. FEIGELSTOCK, The nilstufe of rank two torsion-free groups, Acta Sci. Math., 36 (1974), 29—32. [2] L. FUCHS, Infinite Abelian Groups, Vol. II, Academic Press (New York, 1973). [3] B. J. GARDNER. Rings on completely decomposable torsion-free abelian groups, Comment. Math. Univ. Carolinae, IS (1974), 381—392. [4] R. REE and R. J. WISNER, A note on torsion-free nil groups, Proc. Amer. Math. Soc., 7 (1956), 6— 8 . [5] A. E. STRATTON and M. C. WEBB, Type-sets and nilpotent multiplications, Acta Sci. Math., 41 (1979), 209—213. [6] T. SZELE, Gruppentheoretische Beziehungen bei gewissen Ringkonstruktionen, Math. Z., 54 (1951), 168—180. [7] C. VINSONHALER and W. J. WICKLESS, Completely decomposable groups which admit only nilpotent multiplications, Pac. J. Math., 53 (1974), 273—280. [8] M. C. WEBB, A bound for the nilstufe of a group, Acta Sci. Math., 39 (1977), 185—188. [9] W. J. WICKLESS, Abelian groups which admit only nilpotent multiplications, Pac. J. Math., 40 (1972), 251—259. UNIVERSITY OF GHANA P.O. BOX 62 LEGON, GHANA 17 (5 

Nilpotent torsion-free rings and triangles of types

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Nilpotent torsion-free rings and triangles of types

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