Connection Between Bounded Distributive Lattices and Module Theory: Applications Linking Both Areas?

I usually work with lattices, and one of my professors specializes in this topic as well. However, in the course I am taking, we focus on classical concepts from module theory, rings, and fields, including the Jacobson radical. This has led me to wonder whether there is a non-trivial connection between these two fields—namely, whether lattice-theoretic results or structures can be applied in module theory, or vice versa.

In particular, I have observed that if we consider a bounded distributive lattice $(L, \vee, \wedge, 0,1)$ and interpret it as an idempotent semiring (where the "addition" is $ \vee $ and the "multiplication" is $ \wedge $), the idempotency property implies that for every $ a \in L $,
$$ a \vee a = a. $$ Consequently, no element $ a \neq 0 $ is nilpotent. This leads to the triviality of the Jacobson radical, defined as the intersection of all maximal ideals, meaning that
$$ J(L) = \{0\}. $$

My questions are as follows:

1. Is there any existing study or result in the literature that non-trivially connects lattice theory (or idempotent semirings) with module theory? Specifically, I am interested in any application or categorical equivalence that allows translating or reinterpreting classical concepts—such as the Jacobson radical or other radicals—between both theories.

2. Since the Jacobson radical is trivial in idempotent semirings, have other types of radicals (or analogous notions) been defined or studied in this setting that might be of greater interest?

3. Additionally, I would like to know about concrete examples where lattices are applied to modules or, conversely, where modular concepts are used to address problems in lattice theory. I should clarify that, while I am aware of tropical algebras, that is not the line of application I am interested in.

I would greatly appreciate any references, articles, or prior work that help bridge these two areas. My goal is to explore an exercise from my course by relating it to semirings (and, in particular, lattices) to provide a fresh perspective in my presentation.