Spectra of the \(\Gamma\)-invariant of uniform modules (Q5946462)

The \(\Gamma \)-invariant of the title takes values in the Boolean algebra \(B(\kappa)\) consisting of all subsets of a regular uncountable cardinal \(\kappa\) modulo the filter of subsets containing a closed unbounded set. It is defined for those uniform modules which have a cofinal strictly decreasing continuous chain of length \(\kappa\) of non-zero submodules; such modules are called strongly uniform of dimension \(\kappa\). The invariant \(\Gamma(M)\) may be regarded as measuring the failure of the submodule lattice of \(M\) to be relatively complemented. These notions were introduced and studied in earlier papers of the second author [see Algebra Log. Appl. 9, 471-491 (1997; Zbl 0944.16003), Trends in Mathematics 327-340 (1999; Zbl 0937.16016), and jointly with the reviewer, Algebra Univers. 40, No. 4, 427-445 (1998; Zbl 0936.06008)]. The question arises as to which elements of \(B(\kappa)\) can be realized as the \(\Gamma\)-invariant of a module over a particular ring, or over a ring in some particular class of rings; the elements realized constitute the spectra of the title. In the earlier papers some partial answers are provided to this question.NEWLINENEWLINENEWLINEIn the present paper the main theorem is that for any field \(F\) and any uncountable cardinal \(\lambda\), there is an \(F\)-algebra \(R\) such that for every regular uncountable \(\kappa\leq\lambda\), every element of \(B(\kappa)\) equals \(\Gamma(M)\) for some \(R\)-module \(M\). It is noted that the algebra \(R\) that is constructed is not von Neumann regular. It is also noted that for any ring \(R\), there is an upper bound to the dimension of strongly uniform modules, and hence to the cardinals \(\kappa\) to which the \(\Gamma\)-invariants of \(R\)-modules belong. Later sections of the paper deal with consequences for the \(\Gamma\)-invariant of two-sided ideal lattices and for the structure of Ziegler spectra.