Let $\mathcal{I}^{\bullet}(\Bbb{Z})$ be the set of nonzero ideals of $\Bbb{Z}$. It is a cancellative multiplicative monoid since $\Bbb{Z}$ is a PID. Define $\mathcal{I}^{\bullet} \xrightarrow{\phi} \Bbb{Z} : n\Bbb{Z} \mapsto n$. Then $\phi((n)(m)) = mn$ and $\phi((m) + (n)) = \gcd(m,n)$. Thus at least $\phi$ is a multiplicative monoid hom.
The reason for looking at this map is that the twin prime conjecture addresses the generators and not the ideals they generate. But this map links the two.
The inverse map $\psi : \Bbb{Z} \to \mathcal{I}^{\bullet} : n \to n \Bbb{Z}$.
The two maps together form an antitone galois connection, where the poset on $\mathcal{I}^{\bullet}$ is subset inclusion and the poset on $\Bbb{Z}$ is divisibility. See Pete L. Clark's CA book, pg. 32.
Is this enough to make an ideal form of the twin prime conjecture?