Radicals commuting with Cartesian products
Given an Abelian group \(G\), the group radical is defined by \(R_G(X)=\bigcap\{\text{Ker }\phi\mid\phi\colon X\to G\}\), for Abelian groups \(X\). This radical does not always commute with infinite direct products (for instance, when \(G=\mathbb{Q}\), it turns into the torsion radical). Given any radical \(R\) in the category of Abelian groups, its norm \(\|R\|\) is defined to be the least cardinal \(\kappa\) for which \(R\) does not always commute with \(\kappa\)-direct products of groups. If this cardinal exists it has to be regular. Assuming GCH, if \(\kappa\) is any uncountable regular cardinal with the property that no cardinal \(<\kappa\) is weakly compact, then there exists a torsion-free group \(G\) of cardinality \(2^\kappa\), with \(\|R\|=\kappa\). The coslender radical is a non-trivial example of a radical with norm equal to the first non-measurable cardinal [see \textit{R. Dimitrić}, Proc. Conf., Hobart/Aust. 1987, Pitman Res. Notes Math. Ser. 204, 41-50 (1989; Zbl 0671.20053)].
radicals commuting with infinite products
scientific article; zbMATH DE number 1309628