Prime Ideal in UFD will be the filter in the lattice derived from the UFD

We know that the concept ideal in the lattice is generalised from $\mathrm{Ring}$$\mathrm{Ring}$.

The initial idea is Boolean Algebra (which is a kind of nice lattice) is equivalence to a Boolean Ring, and we could define an ideal in a Boolean Ring, thus we get the ideal in Boolean Algebra.

Consider the classical example:

$$\begin{array}{}\text{(1)}& (P(S),\mathrm{\Delta },\cap )\end{array}$$

is a ring, an ideal in ring theory should have the following properties.

$†.I$$\dagger. I$ is an abelian group.

$†.\mathrm{\forall }r\in R,a\in I,ra\in I$$\dagger.\forall r\in R,a\in I,ra\in I$.

If you consider $(P(S),\mathrm{\Delta },\cap )$$(P(S),\Delta,\cap)$,

Then $a\mathrm{\Delta }b\in I$$a\Delta b\in I$, and $\mathrm{\forall }x\in R,x\cap a\in I⟺x\subseteq a$$\forall x\in R, x\cap a \in I\iff x\subseteq a$, thus $I$$I$ is closed under $\wedge$$\wedge$ as well.

Thus $I$$I$ is closed under $\vee$$\vee$, since $a\vee b=a\mathrm{\Delta }b\mathrm{\Delta }(a\cap b)$$a\vee b=a\Delta b\Delta (a\cap b)$.

That is why In lattice, the definition of ideal is

$$\begin{array}{}\text{(2)}& \mathrm{\forall }a,b\in I,a\vee b\in I\end{array}$$

$$\begin{array}{}\text{(3)}& \text{if }x\le a\in I,\text{then }x\in I.\end{array}$$

Then the filter is the dual of ideal.

i.e.

$$\begin{array}{}\text{(4)}& \mathrm{\forall }a,b\in F,a\wedge b\in F\end{array}$$

$$\begin{array}{}\text{(5)}& \text{if }x\ge a\in F,\end{array}$$

But a pretty funny thing is some ideals in some kind of ring form a filter.

For example, Let us consider the initial object $\mathbb{Z}$$\mathbb Z$.

Then the prime ideal $p\mathbb{Z}$$p\mathbb Z$ is a filter in $(\mathbb{N},gcd,\mathrm{lcm})$$(\mathbb N,\gcd,\mathrm{lcm})$.

Since if $a,b\in p\mathbb{Z}$$a,b\in p\mathbb Z$, then $gcd(a,b)\in p\mathbb{Z}$$\gcd(a,b)\in p\mathbb Z$, and if $x\ge a⟺\mathrm{lcm}(x,a)=x$$x\ge a\iff\mathrm{lcm}(x,a)=x$ thus $p|x$$p| x$.

In general, Let $R$$R$ be a $\mathrm{U}\mathrm{F}\mathrm{D}$$\rm UFD$, and $a\sim b⟺a=rb$$a\sim b\iff a=rb$, where $r$$r$ is a unit. Then $(R/\sim ,gcd,\mathrm{l}\mathrm{c}\mathrm{m})$$(R/\sim,\gcd,\rm lcm)$ form a lattice.

Then $(p)$$(p)$ as an ideal in $R$$R$ and a filter in $(R/\sim ,gcd,\mathrm{l}\mathrm{c}\mathrm{m})$$(R/\sim,\gcd,\rm lcm)$.