Is it possible to classify all the primary ideals of the polynomial ring $K[X_1,\ldots ,X_n]$ where $K$ is a field.
Or, give a big class of examples of primary ideals which are not prime ideals.
Is it possible to classify all the primary ideals of the polynomial ring $K[X_1,\ldots ,X_n]$ where $K$ is a field. Or, give a big class of examples of primary ideals which are not prime ideals. 


When you restrict to special classes like monomial or binomial ideals (those generated by polynomials with one (monomial) or two (binomial) terms) then combinatorial characterizations exist. For instance, a monomial ideal $I\subset K[X_1,\dots,X_n] =:S $ is primary if and only if in the quotient $S/I$ every image of a variable is either regular or nilpotent. For binomial ideals the story is more complicated but things can be said. See Eisenbud/Sturmfels "Binomial ideals", Dickenstein/Matusevich/Miller, "Combinatorics of binomial primary decomposition", and Kahle/Miller "Decompositions of commutative monoid congruences and binomial ideals". 


Symbolic powers give big classes of primary ideals which are not prime. Recall that given a prime $P$, the $n$th symbolic power of $P$ (denoted by $P^{(n)}$) is the $P$primary ideal in the minimal prime decomposition $P^n$. It can also be described as $(P^n R_P) \cap R$. It is perhaps surprising that $P^{(n)} \neq P^n$, but the only real large class of ideals where that always happens is for prime ideals which are cut out by regular sequences (ie, complete intersections). In algebraic geometry, $P$primary ideals show up frequently when $P$ is a height one ideal. Indeed, suppose that $D$ is a prime divisor on a normal variety $X = \text{Spec} R$. Then $$\Gamma(X, \mathcal{O}_X(nD)) = P^{(n)}$$ where $P$ is the prime ideal defining $D$. For example, consider the prime $P = (x,y) \subseteq k[x,y,z]/(x^2  yz)$, it is a good exercise to verify that $P^{(2)} \neq P^2$ in this example. A poor classification: Based upon the symbolic power idea, one can classify $P$primary ideals as follows (for any prime $P$). The set of $P$primary ideals is equal to the set of all $$ Q \cap R $$ where $Q$ runs over all ideals of $R_P$ such that $\sqrt{Q} = P R_P$. 

