Necessary conditions for the coperiodicity of quotients of direct products of Abelian groups (Q762275)

An (abelian) group A is called coperiodic iff \(Ext(G,A)=0\) for any torsion-free group G. Let \(\{A_ i\}_{i\in I}\) be a family of groups, let \(I_ 1\) be the set of \(i\in I\) such that \(A_ i\) is not coperiodic and let \(\oplus_ KA_ i\) denote the K-direct sum (i.e. the subgroup of the direct product \(\prod A_ i\) consisting of all elements with support belonging to the ideal K of the algebra B(I) of all subsets of I). Main results of the paper are: 1. If a factor group \(\oplus_ KA_ i/T\) is coperiodic then \(B(I_ 1)\cap K\) is contained in the \(\sigma\)-ideal of B(I) generated by supports of elements in T. 2. If \(K_ 1\subseteq K_ 2\) are ideals of B(I) and \(K_ 2\) is a \(\sigma\)-ideal then the factor group \(\oplus_{K_ 2}A_ i/\oplus_{K_ 1}A_ i\) is coperiodic iff \(B(I_ 1)\cap K_ 2\) is contained in the \(\sigma\)-ideal generated by \(K_ 1\). 3. If \(K_ 1\) is an ideal of B(I) then there exists a maximal element in the collection of all ideals K such \(K_ 1\subseteq K\subseteq B(I)\) and the group \(\oplus_ KA_ i/\oplus_{K_ 1}A_ i\) is coperiodic.