## Endomorphism Rings of Modules Whose Cardinality Is Cofinal to Omega

We want to consider torsion-free R-modules over a ring R. In Section 3 the ring R will be a principal ideal domain and in Section 4 we allow more general commutative rings R. However generally we assume that R has a distinguished countable, multiplicatively closed subset S of non-zero divisors. We also may assume that 1 ∈ S and say that an R-module G is torsion-free if gs = 0 (g ∈ G, s ∈ S) only holds if g = 0. Moreover, G is reduced (for S) if ∩ s ∈ S G s = 0 . Throughout we suppose that R is reduced and torsion-free (for S). The reader will observe that under these restrictions two kinds of realization theorems for R-algebras A as endomorphism algebras of suitable modules G are known. If we are lucky, then we find an R-module G with End R G = A . ( STRONG )