On the Hasse diagram for ideals

Question

When consulting the wikipedia regarding prime ideals, the following Hasse diagram (is it also a lattice?) is provided as representation:

https://en.wikipedia.org/wiki/Prime_ideal

Any idea of who first made use of that visualization? References are welcome. Any advantages to it? Here I find this other representation:

http://mathworld.wolfram.com/Ideal.html

Thanks in advance.

Answer 1

Any idea of who first made use of that visualization?

Anyone who knew about posets could produce such a graph, since it is just a sublattice of the power set of the ring. The ideas are so common it does not really seem possible to attribute primacy to any one person.

In another sense, the graph ( or at least its reverse order) has been used since antiquity since it just re-expresses divisibility (see this for example. )

Any advantages to it?

As with any Hasse-like diagram, with sufficient knowledge of the graph you can tell the order relation between any two elements.

Is it also a lattice?

Sure: In terms of the containment order it is a poset in which $(a)\wedge(b)=(\mathrm{lcm} (a,b))$ and $(a)\vee (b)=(\gcd (a,b))$. That is an order theoretic lattice.

You shouldn't confuse this with the 'lattice' used in the other example you gave, because that is a lattice in another sense. In the divisibility order the the two operators are reversed.