Modules with absolute endomorphism rings.
The authors construct absolutely fully rigid families of modules of any size smaller than the first Erdős cardinal. Roughly, a fully rigid system of modules is a system of submodules with prescribed endomorphism algebras and only the minimally necessary homomorphisms between each other (i.e. composition of inclusion and endomorphisms). The new feature of the result is that the construction is absolute, i.e. the families remain fully rigid and will not have new homomorphisms between its members in all generic extensions of the universe. The first Erdős cardinal is known to be a bound of the size of absolutely rigid mathematical objects of many kinds, so the size restriction is necessary. The underlying set theoretic results are established by \textit{P. C. Eklof} and \textit{S. Shelah} [in: Abelian groups and modules. Proc. int. conf. Dublin, Ireland, 1998. Basel: Birkhäuser. Trends in Mathematics. 257-268 (1999; Zbl 0952.20044)] using \textit{S. Shelah} [Isr. J. Math. 42, 177-226 (1982; Zbl 0499.03040)]. As an application, absolutely fully rigid families of modules with countably many distinguished submodules were constructed by \textit{R. Göbel} and \textit{S. Shelah} [in Proc. Am. Math. Soc. 135, No. 6, 1641-1649 (2007; Zbl 1122.13006)]. A homomorphism between two such modules is a module homomorphism mapping every distinguished submodule into the corresponding distinguished submodule. The present article continues this process exactly as has been done in the non-absolute case. So it is shown that five distinguished submodules are sufficient, and then this is applied to show that an absolutely fully rigid system of modules (with no distinguished submodule) exists over many rings satisfying some mild condition.
scientific article; zbMATH DE number 5508702
scientific article