Vol. 54, No. 1, 1974

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The module structure of Ext (F, T) over the endomorphism ring of T

Samir A. Khabbaz and Elias Hanna Toubassi

Vol. 54 (1974), No. 1, 169–176

§0. Introduction. Seotion 1 of the paper deals with the structure of the left End(T)-module Ext(F,T) where F is a torsion-free group and T a torsion group. The main theorem states that if the rank of F is n then the rank of Ext(F,T) over End (T) is at most n. It is also shown that Ext(F,T) is a torsion End (T)-module.

In §2 the case when F is a torsion-free divisible group of finite rank is considered. It is shown that m, the elements of the End (T)-End (F) bimodule Ext(F,T) whose splitting length is at most m, are a sub-bimodule. Moreover = i=1i as well as m are not finitely generated sub-bimodules. However, when T is the direct sum of cyclic groups, every finite number of elements of can be embedded in a submodule generated by at most rank F elements belonging to m.

We introduoe some notation. The word “group” will always mean abelian group. Q will stand for the additive group of rational numbers, N the natural numbers, and N0 the nonnegative integers. Let G be a group, then T(G) is the maximal torsion subgroup of G. For x G then hp(x) will denote its generalized p-height in G,o(x) its order, and xthe cyclio subgroup generated by x. For a torsionfree element x G,U(x), the Ulm matrix of x, will refer to the matrix hp(pix) over all i N0 and primes p[9]. For the module structure of Ext(F,T) over End (T)-End (F) we refer the reader to [8]. Following [8] for α End(T)[γ End(F)] and E Ext(F,T) by αE[] we mean the extension arising from the pushout [pullback] diagram of E along α[γ].

Mathematical Subject Classification 2000
Primary: 20K40
Received: 7 March 1973
Revised: 18 September 1973
Published: 1 September 1974
Samir A. Khabbaz
Elias Hanna Toubassi