Modules are distinguished from commutative rings by the fact they are defined by action by an external set. Recall, that commutative groups form an abelian category so for any commutative group G a ring action on G can be defined by a ring homomorphism to the endomorphism ring End(G). The scalar multiplication action also forms a group endomorphism, which makes modules a special case of groups with operators. The transition to the submodules lattice Sub(M) proceeds in a similar manner.
Lattices with operators:
Let L be a lattice, then a lattice with operators can be defined by an indexed family of endomorphisms in the category of preorders and monotone maps. This clearly forms an abstract class of ordered algebraic structures like residuated lattices and quantales. We can make R-submodules into a lattice with operators in the standard manner:
\cdot : Ideals(R) \times Sub(M) \to Sub(M)
The only thing that needs to be proven really is that ideal action on submodules is indeed monotone. Let I be a fixed ideal and suppose that M_1 \subseteq M_2. Let b be an element of IM_1 then b = \sum a_i x_i for some a \in I and x_i \in M_1. Now by the fact that M_1 \subseteq M_2 this means that b \in \sum a_i x_i for some a_i \in I and x_i \in M_2 so b \in IM_2 which implies IM_1 \subseteq IM_2. This confirms that ideal action is monotone. Therefore, we can distinguish between two cases: (1) ideals which form a quantale and (2) submodules which form a lattice with operators.
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