Groups in the class semigroups of valuation domains (Q1817272): Difference between revisions

Given a commutative valuation domain, the class semigroup \(S(R)\) of \(R\) is the factor semigroup of the semigroup of non-zero fractional ideals of \(R\), mod the non-zero principal fractional ideals. It is shown that \(S(R)\) is a Clifford semigroup (disjoint union of groups) possessing some special properties. A characterization of abelian groups that appear as group components of \(S(R)\) is given. These groups are associated with the idempotent non-zero prime ideals of \(R\). An abelian group \(A\) is \(\kappa\)-realizable (for an infinite cardinal \(\kappa)\), if, for some torsion-free abelian group \(G\), \(A\cong\overline G/G\), where \(\overline G\) is the completion of \(G\) in a \(\kappa\) long descending filtration of subgroups (intersecting trivially). For an uncountable regular cardinal \(\kappa\), every abelian group is \(\kappa\)-realizable. A group is \(\omega_0\)-realizable iff it is cotorsion. This notion is further generalized to skeleton-realizable groups. Reviewer's remark: A number of ideas in this paper represent rephrasing of ideas presented by the reviewer in 1989, in Oberwolfach: Cf. \textit{R. M. Dimitrić}, Kobe J. Math. 9, No. 1, 43-61 (1992; Zbl 0783.13010) and Acta Math. Inform. Univ. Ostrav. 1, 13-25 (1993; Zbl 0857.13020)].