Submodule lattices with operators

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.