On a problem of Wenzel on equational compactness of modules (Q1966119)

Let \(R\) be an associative ring with 1 and \(M\) be a left unital \(R\)-module. Denote by \(M^{(I)}\) the direct sum of \(|I |\) copies of \(M\). \(M\) is \(x_0\)-compact if every finitely solvable system of the form \(x_0 + r_jx_j=a_j\) \((j\in J\), \(r_j\in R\), \(a_j \in M)\) is solvable. If \(M^{(I)}\) is \(x_0\)-compact for each set \(I\), \(M\) is called \(\Sigma \)-\(x_0\)-compact. G. H. Wenzel posed the problem whether an arbitrary \(R\)-module is \(\Sigma \)-\(x_0\)-compact. The author gives a negative answer and he proves also the adjacent result that, over a wide class of commutative rings, if every module is \(x_0\)-compact, then the ring has a finite representation type.