I am reading currently Michael Atiyah. I encountered the following proposition. It seems to me that the following condition characterizes prime ideals geometrically that is it tells us how ideal fits between bunch of prime ideals. Does the condition characterizes prime ideals ? That is if ideals satisfy the condition below then it is necessarily prime ?
Conditions:
1) Let $p_1,...,p_n$ be prime ideals and let $a$ be an ideal contained in $\cup p_i$. Then $a \subset p_i$ for some i.
2) Let $a_1,...,a_n$ be ideals and let $p$ be a prime ideal containing $\cap a_i$, then $a_i \subset p$ for some i.
Follow up question: Is there a different description of prime ideals by maybe how it sits among other ideals if the above condition doesn't characterize it ?
does not characterize prime ideals: there are some rings in which the prime avoidance lemma works for nonprime ideals.
does not characterize prime ideals: Let $R$ be a commutative ring with linearly ordered ideals, but is not a field. Denote its maximal ideal by $M$. Suppose additionally that $M\neq M^2$. (Take $F[x]/(x^3)$ where $F$ is a field, for example.)
For any two ideals, $A\cap B=\min(A,B)$ according to the linear order of the ideals. Now, $\cap_{i=1}^nA_i\subseteq M^2$ clearly implies that one of $A_i$ is contained in $M^2$ (since the intersection is equal to one of the $A_i$.) But $M$ is obviously not prime since $M\nsubseteq M^2$.
Note by the same arguments, every ideal in $R$ satisfies the proposed condition, prime or not!