ABSTRACT
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 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187605/d46d5034-f914-41b4-a96f-9f0731bff786/content/eq1783.tif"/> . 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 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187605/d46d5034-f914-41b4-a96f-9f0731bff786/content/eq1784.tif"/>
