Butler groups of infinite rank. II

Abstract

A torsion-free abelian group G is called a Butler group if Bext(G, 7") = 0 for any torsion group T . We show that every Butler group G of cardinality N, is a ¿?2"8rouP; 'e-' C7 is a union of a smooth ascending chain of pure subgroups Ga where Ga+X = Ga + Ba , Ba a Butler group of finite rank. Assuming the validity of the continuum hypothesis (CH), we show that every Butler group of cardinality not exceeding N^ isa ß2-group. Moreover, we are able to prove that the derived functor Bext (A , T) = 0 for any torsion group T and any torsion-free A with \A\ < Kw . This implies that under CH all balanced subgroups of completely decomposable groups of cardinality < Hm are 52"ßrouPs-(1) Bext(t7, T) = 0 for all torsion groups T.(2) Bext(C7, T) = 0 for all 1,-cyclic torsion groups T.(3) G is a B2-group.(ß) The derived function Bext (G, T) = 0 for any torsion group T.

Rings whose additive endomorphisms are ring endomorphisms

“…Define a subgroup H of the external direct sum 0 a < 2 « -^2 as follows: if ep denotes the vector with 1 in the beta-th coordinate and zeros elsewhere, we let Q a B a e a + £ Q< 2 >(e a -e 0 ) + Q< s >( ei -e 0 ). Let a = <(Q (2) ) denote the type of Q< 2 >, let T = t(Q ( s ) ) and let r a = <(Q a ). One verifies the following.…”

“…Also, since only finitely many components of w are nonzero, it is possible to write w such that n a = np = n £ N for all a and /3. By hypothesis, given any m £ N , there exists y m 6 H such that 2 m y m -w, and with r;,(™ G Q Q , C € Q ( 2 ) , <tf* = 0, E C = 0, P? € Q ( 3 ) , p?…”

“…Let x and y be elements of R such that xy $. 2 U R. Then xy £ R [2] has finite 2-height n -1 ^ 0. If xy = 2 n~1 c for some c £ R then c has order 2 n and [4, p.117, 27.1] implies R + = (c) @ W. Hence R = (c)-f W. Let p and q be integers and v,w £ W such that x = pc + v and y = qc + w. Then 2 n~1 c -xy = pqc 2 + vw which implies pq is odd and c 2 = 2 n -1 c .…”

“…Not every subdirectly irreducible homomorphic image of an AE-ring is an AE-ring.PROOF: Let G be the group of 3.3. Then there exists x e G and a subgroup A of G such that tG + 2G < A and G/A = {x + A) ~ Z(2). By 3.2, there exists an AE-ring R on G with x 2 = a and R • A -0 = A • R. Let B be a basic subgroup of P such that P/B ~ Z{2°°).…”

“…By 3.2, there exists an AE-ring R on G with x 2 = a and R • A -0 = A • R. Let B be a basic subgroup of P such that P/B ~ Z{2°°). Then (P/B)[2] = (a + B). Since x £ A, the order of x must be infinite so that (x + B) t~\ P/B = 0.…”