A ring ideal can be characterized by the two rules: $$(a\in I) \wedge (a ~ \textrm{divides} ~ b) \implies b \in I$$ $$ a,b \in I \implies \textrm{gcd}(a,b) \in I$$ (the usual definition states $a,b \in \implies a+b \in I$ instead of gcd, but it's equivalent by the Bezout's identity)
When we use the notation $(b \le a) :\Leftrightarrow (a ~ \textrm{divides} ~ b)$ and thus also consistently $a \vee b := \textrm{gcd}(a,b)$, we get exactly the definition of an ideal in a poset.
However, the usual convention is the opposite, using $(a \le b) :\Leftrightarrow (a ~ \textrm{divides} ~ b)$ and thus also $a \wedge b := \textrm{gcd}(a,b)$. And in this case we obtain the dual of an ideal, called a filter.
Then my question is, why we don't call ring ideals "ring filters" instead? Isn't it unconsistent to call them ideals? Shouldn't the poset versions of filter and ideal definitions be interchanged in order to get back the consistency?
Edit: Arnaud D. found what is wrong. My definition of an ideal is not equivalent to the original one. Indeed, there are ideals that are not principal ideals.