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Modules with two distinguished submodules. (English) Zbl 0808.16030

Fuchs, Laszlo (ed.) et al., Abelian groups. Proceedings of the 1991 Curaçao conference. New York: Marcel Dekker, Inc.. Lect. Notes Pure Appl. Math. 146, 97-104 (1993).
Let \(R\) be a commutative ring with \(1 \neq 0\) and \(S\) be a fixed multiplicatively closed subset of \(R\) containing no zero-divisors, \(1 \in S\). An \(R\)-module \(G\) is \(S\)-reduced if \(\bigcap_{s \in S} sG = 0\) and \(S\)-torsionfree if \(ms = 0\) implies \(m = 0\). Let \(I\) be a partially ordered set, \(I = (I,\leq)\) and \(\lambda\) an infinite cardinal. A sequence \({\mathbf G} = (G,G^ i : i \in I)\) consisting of an \(R\)-module \(G\) and distinguished submodules \(G^ i\) with \(G^ i \subseteq G^ j\) for all \(i \leq j\) is called an \(R_ I\)-module. The main result of the paper is the following Theorem. If \(A\) is any \(R\)-algebra with \(A_ R\) an \(S\)-reduced, \(S\)-torsionfree \(R\)-module and \(| A| \leq \lambda\), then there exists an \(R_ 2\)-module \({\mathbf G} = (G, G^ 0, G^ 1)\) such that \(| G | = \lambda^{\aleph_ 0}\) and \(\text{End }{\mathbf G} \cong A\). Corollary. Let \(I = (I,\leq)\) be a partially ordered set and \(\lambda\) an infinite cardinal. The following are equivalent. (1) \(I\) is not linearly ordered. (2) For all \(R\)-algebras \(A\) over a commutative ring \(R\) with a multiplicatively closed set \(S\) such that \(A_ R\) is \(S\)-reduced and \(S\)-torsionfree of cardinality \(\leq \lambda\) there exists a family \({\mathbf G}_ \alpha\) \((\alpha \subseteq \lambda)\) of \(S\)-reduced and \(S\)-torsionfree \(R_ I\)-modules of rank \(\lambda^{\aleph_ 0}\) with \(\text{Hom}({\mathbf G}_ \alpha, {\mathbf G}_ \beta) = A\) if \(\alpha \subseteq \beta\) and \(\text{Hom}({\mathbf G}_ \alpha, {\mathbf G}_ \beta) = 0\) if \(\alpha \nsubseteq \beta\).
For the entire collection see [Zbl 0778.00023].

MSC:

16S50 Endomorphism rings; matrix rings
06A06 Partial orders, general
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
16D80 Other classes of modules and ideals in associative algebras
16G20 Representations of quivers and partially ordered sets
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