Does there exist some good notion of "filters" for arbitrary associative rings with unity that would generalize filters of Boolean rings?
For me, if $R$ is an associative ring with unity $1$, it would seem natural to define a filter of $R$ as any subset of the form $1 + I$, where $I$ is a (two-sided) ideal of $R$.
(This question is in a sense a follow-up to that one.)