Square-free modules with the exchange property.
A right \(A\)-module \(M_A\) has the `\(\aleph\)-exchange property', for some cardinal \(\aleph\geq 2\), if whenever there are right \(A\)-module decompositions \(A=M\oplus N=\bigoplus_{i\in\aleph}A_i\) then there exist submodules \(A_i'\subseteq A_i\), for each \(i\in\aleph\), so that \(A=M\oplus(\bigoplus_{i\in\aleph}A_i')\). If \(M_A\) has the \(\aleph\)-exchange property for all cardinals \(\aleph\) (respectively, all finite cardinals), then we say \(M_A\) has the `full exchange property' (respectively, the `finite exchange property'). A module is called `square-free' if there is no nonzero submodule isomorphic to a square \(X\oplus X\). The main result of the paper is the one addressing its title, Theorem 9: ``If \(M\) is a square-free module with the finite exchange property then \(M\) has the full exchange property''. Actually the author proves a result in more generality that deals with topological conditions on a ring, which are the following: a ring \(R\) has a `left linear Hausdorff' topology if there is a ring topology with a basis of neighborhoods of zero, \(\mathcal U\), consisting of left ideals, satisfying \(\bigcap_{U\in\mathcal U}U=(0)\). A collection \(\{x_i\}_{i\in I}\) of elements of \(R\) is `summable' to \(r\in R\) if for every \(U\in\mathcal U\) there is a finite set \(F'\subseteq I\) (depending on \(U\)) such that \((\sum_{i\in F}x_i)-r\in U\) for all finite sets \(F\supseteq F'\). A family of elements is `\(\Sigma\)-Cauchy' if for every \(U\in\mathcal U\) there is a finite set \(F_U\subseteq I\) so that \(x_i\in U\) when \(i\not\in F_U\). Finally, a ring \(R\) is `\(\Sigma\)-complete' if all \(\Sigma\)-Cauchy families are summable. Also, a ring \(A\) is said to be `Abelian' if all idempotents are central, and a module is `Abelian' when its endomorphism ring is an Abelian ring. Thus, the more general aforementioned result, Theorem 8, reads: ``Let \(R\) be an exchange ring with a left linear, \(\Sigma\)-complete, Hausdorff topology. If \(R/J(R)\) is Abelian, then \(R\) is a full exchange ring''. The paper finishes by showing that square-free modules with exchange satisfy an infinite version of the \((C_3)\) property (a module has \((C_3)\) if the sum of two non-intersecting summands is again a summand).
scientific article; zbMATH DE number 5696933
scientific article