Representations of distributive algebraic lattices by lattices of two-sided ideals of rings, resp. submodule lattices of modules

Pavel Růžička

Charles University, Prague

We will present some results concerning representations of algebraic lattices in ideal lattices of rings, resp. submodule lattices of modules. We prove that any algebraic lattice which can be represented as the lattice of two-sided ideals of some ring can be represented as the submodule lattice of some module as well. Then we focus on algebraic distributive lattices. An algebraic distributive lattice is determined by the semilattice of its compact elements and these semilattices correspond to distributive $\left\langle \vee,0 \right\rangle$-semilattices. We present the following results: