One of the problems that has been a roadblock in my understanding of ideals has been how one would find them. One way of finding an I of some ring R would be to say $ \forall x \in I, \forall r \in R : x \cdot r \in I $ This however means that we would have to "guess" at an ideal I and then see if each element in I is indeed an ideal. Is this correct? If I am right, without guessing we could keep track of all the cosets of each $R/I$ and then find unions and intersections, and a maximal ideal would correspond to ? - the union of some amounts of cosets of "ideal elements"? - How does this work?
It is this "guessing" that makes me wonder whether this is an NP problem - but if we didn't guess would we have some sort of P-complexity computation of each individual ideal and the union of them all being the maximal ideal? - I don't remember the particular laws of ideals so please correct me.
--Edit-- In respect to Conifolds questions:
I want to find if an element is in any non-trivial ideal I i.e. $I \neq R$.
ay someone hands me a set $S \subset R$ and wants to know if it is an ideal of R, what kind of algorithm would tell me - can we do better than multiplying every element of S by every element in R and checking to see if it is in S? I suppose that I don't see much of why one would need to "guess" anymore as instead one could take each element $s$ and multiply it by all other elements $r$ of the ring $R$ and then say it is possibly an ideal $S$ if all the other $r \cdot s = t$ such that $t \in S$. Of course then each of these elements multiplied by each element $r$ in the ring would ultimately have to be in the ideal also - and so on. This seems like a large amount of dependencies and thus computations. It seems to me that knowing whether a ring is finitely generated would help on complexity constraints, but in the worst case scenario (which would be no constraints)?