Ideal class tables

The classes of ideals consisting of prime ideals, primary ideals, and ideals with prime radical are all intimately related to one another: in fact they form a totally ordered hierarchy of classes. To determine the different types of classes of ideals that can be formed, I find it interesting to form membership tables relating to products of pairs of elements $ab \in I$.

Prime ideals:

	$a \in I$	$a \in \sqrt{I}$	$a \not\in \sqrt{I}$
$b \in I$
$b \in \sqrt{I}$
$b \not\in \sqrt{I}$

Primary ideals:

	$a \in I$	$a \in \sqrt{I}$	$a \not\in \sqrt{I}$
$b \in I$
$b \in \sqrt{I}$
$b \not\in \sqrt{I}$

Ideals with prime radical:

	$a \in I$	$a \in \sqrt{I}$	$a \not\in \sqrt{I}$
$b \in I$
$b \in \sqrt{I}$
$b \not\in \sqrt{I}$

We see that these are indeed the three types of ideals that are formed by membership conditions on products related to the ideal and its radical. The nice thing about this table is it describes primary ideals in a more commutative manner then the standard definition $a \not\in I \Rightarrow b^n \in I$. From this implication based definition, you might not realize that an ideal is primary iff $ab \in I$ implies that $a \in I \lor b \in I \lor (a \in \sqrt{I} \land b \in \sqrt{I})$. These tables display that fact visually.