Endomorphism algebras of modules with distinguished torsion-free elements (Q1904079)

Let \(E(M)\) denote the algebra of \(R\)-linear endomorphisms of a unitary \(R\)-module \(M\), \(R\) being a fixed discrete valuation ring. The author contributes to the complex of problems, which arose from Kaplansky's result, that every isomorphism \(E(M)\to E(N)\) is induced by an isomorphism of the modules themselves if both \(M\) and \(N\) are torsion. He introduces the concept of a stable element \(x\) of a mixed module \(M\), that is an element for which (i) there exists \(\delta\in E(M)\) such that \(\delta(x)=x\) and \(\delta M/\langle x\rangle\) is torsion, and (ii) \(M/E(M)x\) is torsion. The two main results then read as follows: (1) Assume \(M\) and \(N\) are reduced modules and the torsion of \(M\) is totally projective. If each module possesses a stable element, then the implication of Kaplansky's result holds. (2) This is true also if only \(M\) possesses a stable element under the additional condition \(\text{rk} (N)\leq \text{rk} (M)\), and both being finite.