Minimal non-totally minimal topological rings
Let \(R\) be a discrete ring and \(R^*\) the group of characters of its additive group. The group \(R^*\) can be regarded in a natural way with a structure of a topological \((R,R)\)-bimodule. The topological group \(R\times R^*\times(\mathbb{R}/\mathbb{Z})\) with the multiplication \((r_1,\xi_1,s_1)(r_2,\xi_2,s_2)=(r_1r_2,r_1\xi_2+\xi_1r_2,\xi_1(r_2)+\xi_2(r_1))\), where \(r_1,r_2\in R\), \(\xi_1,\xi_2\in R^*\), \(s_1,s_2\in\mathbb{R}/\mathbb{Z}\) becomes a locally compact associative ring (denoted by \(M(R)\)). The main result of the paper states that the topological ring \(M(R)\) is minimal. This result implies that there exists a locally compact minimal ring \(R\) with identity having a closed ideal \(I\) such that the factor ring \(R/I\) is not minimal. The last example gives a positive answer to a question posed by D. Dikranjan.
scientific article; zbMATH DE number 1089771
scientific article