Boundedness in locally compact rings (Q1313934)

Let \(R\) be a topological ring and \(M\) a topological \((R,R)\)-bimodule. If on the product \(R\times M\) of the topological groups \(R(+)\) and \(M(+)\) the multiplication is defined by \((r,m)\cdot(r',m')= (r\cdot r',m\cdot r'+ r\cdot m')\), then \(R\times M\) becomes a topological ring. If \(R\) is a locally compact ring, the group \(R^*\) of characters of the group \(R(+)\) is a topological \((R,R)\)-bimodule in a natural way, therefore the ring \(R\times R^*\) can be considered. In the paper is studied the question when for every topological \((R,R)\)-bimodule \(M\) (and, in particular, for a locally compact ring \(R\) for the \((R,R)\)-bimodule \(M = R^*\)) the topological ring \(R\times M\) is left bounded, right bounded.