Note that throughout rings have a multiplicative identity and are not necessarily commutative
Definition: Let $R$ be a ring and let $M\subseteq R$. Then, $M$ is an m-system iff for every $x,y\in M$ there is some $f\in R$ such that $xfy\in M$.
Definition: Let $R$ be a ring and let $N\subseteq R$. Then, $N$ is an n-system iff for every $x\in N$ there is some $f\in R$ such that $xfx\in N$.
For what it's worth, these definitions are designed so that an ideal is prime (resp. radical) iff its complement is an m-system (resp. n-system). (These terms actually come from Lam's book on noncommutative rings, where he uses the term "semiprime" instead of "radical".)
In the case $R$ carries a topology, I would like to prove the following.
Conjecture: Let $R$ be a topological ring, let $N\subseteq R$ be an open n-system, and let $f\in N$. Then, there is an open m-system $f\in M\subseteq N$.
This is a generalization of Lam's Lemma 10.10 and the reason I care about it is because I would like to show that the smallest closed radical ideal containing a closed ideal $I$ is the intersection of all closed prime ideals containing $I$.
Unlike most of Lam's results in this section, the solution here does not seem to be a straightforward generalization of his arguments, and I'm beginning to suspect it might not even be true.
Any ideas on how best to proceed?