The weights of closed subrings of a locally compact ring. (Q390816)

The author proves that for every locally compact infinite ring \(R\) (with weight \(\omega(R)\)) and for every infinite cardinal \(m<\omega(R)\), there exists a closed subring of \(R\) with weight equal to \(m\). He also shows that every infinite compact nilring \(R\) contains an infinite ideal \(I\), which is nilpotent, i.e., \(I^2=\{0_R\}\), where \(0_R\) stands for the zero element of \(R\). The author also provides an example of a compact nilring having infinite metrizable nilpotent ideals.