Submodule lattice quasivarieties and exact embedding functors for rings with prime power characteristic (Q1918966)

The authors study quasivarieties \({\mathcal L}(R)\) of lattices generated by lattices of submodules of \(R\)-modules for rings \(R\) with prime power characteristic \(p^k\). An algebra of expressions \(d\) not dependend on \(R\) is developed, such that each such \(d\) uniquely determines a two-sided ideal \(d_R\) of \(R\). It is shown that \({\mathcal L}(R)\subseteq{\mathcal L}(S)\) makes all implications of the form \(d_S=S\Rightarrow d_R=R\) true, for any such expression \(d\). Further the ordered set \({\mathcal W}(p^k)\) of all lattice quasivarieties \({\mathcal L}(R)\), \(R\) having characteristic \(p^k\), is investigated. For \(k\geq 2\), \({\mathcal W}(p^k)\) has a subset which is order isomorphic to the Boolean algebra of all subsets of a denumerably infinite set, i.e. \({\mathcal W}(p^k)\) is shown to be large and complicated, with ascending and descending chains and antichains having continuously many elements.