A note on semilocal rings

Abstract:

If $(R,{m_1}, \ldots ,{m_w})$ is a semilocal ring whose ideal lattice is topologically complete, it is shown that: given any natural number $n$ and any decreasing sequence $\langle {a_i}\rangle$ of ideals of $R$, there exists a natural number $s\left ( n \right )$ such that ${a_{s(n)}} \subseteq (\bigcap {_i{a_i}) + {m^n}}$ where $m = \bigcap \nolimits _{i = 1}^w {{m_i}}$. This generalizes a well-known theorem on complete semilocal rings.