The notions of a filter and an ideal on a poset make intuitive sense to me, and I can understand why they are dual:
A subset $I\subset P$ of a poset $P$ is an ideal if:
- for all $x\in I$, $y\leq x$ implies $y\in I$
- for all $x,y\in I$ there exists $z\in I$ with $x\leq z$ and $y\leq z$
and a filter is the same thing with all inequalities reversed.
I feel like this should correspond to the notion of a ring ideal, where for a ring $R$ we have $I\subset R$ being a ring ideal if:
- for all $x,y\in I$ we have $x+y\in I$
- for all $x\in I$ and $r\in R$ we have $rx\in I$ and $xr\in I$
but I would like some clarification on this point. Following on, my main question is: is there a corresponding notion of a 'ring filter' which is dual to the notion of a ring ideal in the same way that a filter in a poset is dual to an ideal? Or is there no relation at all except for a coincidence in naming?
The naming isn't a coincidence. An ideal in a Boolean ring is the same thing as an order ideal in the associated Boolean poset, where you can define the order relation by $a \le b$ if $a = ab$. But I don't see any reason to expect a well-behaved associated notion of filter for general rings.