Endomorphism algebras of modules with distinguished partially ordered submodules over commutative rings (Q1181419)

Let \(R\) be a commutative ring with identity and let \(\mathbf I = (I,\leq)\) be a partially ordered set. An \(R_ I\)-module (or \(R\)-representation of \(\mathbf I\)) is a sequence \(\mathbf M = (M, M^ i, i\in I)\) consisting of an \(R\)-module \(M\) and distinguished submodules \(M^ i\) such that \(M^ i\subseteq M^ j\) for all \(i\leq j\in I\). The authors study the endomorphism algebra \(\text{End }\mathbf M = \{\varphi \in \text{End }M\mid M^ i\varphi \subseteq M^ i \text{for all }i\in I\}\). A complete answer is given to the question which partially ordered sets \(I\) allow any prescribed \(R\)-algebra \(A\) to be the endomorphism algebra of some \(R_ I\)-module; they turn out to be the posets of infinite representation type of \textit{M. M. Kleiner} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 28, 42-54 (1972; Zbl 0345.06002)]. The result is extended to `rigid systems' of \(R_ I\)-modules.