Exact embedding functors for module categories and submodule lattice quasivarieties (Q1305451)

For rings with unit \(R\) and \(S\) the relation \(R\precsim S\) means that there exists an exact embedding functor \(F\colon R\text{-Mod}\to S\text{-Mod}\). By \({\mathcal L}(R)\) is denoted the quasi-variety of lattices generated by the family of all submodule lattices \(\text{Su}(_RM)\) for \(M\in R\text{-Mod}\) (a lattice \(L\) is in \({\mathcal L}(R)\) iff it is isomorphic to a sublattice of some \(\text{Su}(_RM)\)). It is known that \(R\precsim S\) iff \({\mathcal L}(R)\subseteq{\mathcal L}(S)\). In this article the lattice structure for submodule lattice quasivarieties is investigated. Let \(\mathcal W\) be the set of all quasivarieties of lattices. For a class of rings \({\mathcal R}'\) by \({\mathcal W}({\mathcal R}')\) is denoted the subset of \(\mathcal W\) consisting of all quasivarieties equal to \({\mathcal L}(R)\) for some \(R\in{\mathcal R}'\). For the class \({\mathcal R}_c\) of all commutative rings it is proved that \({\mathcal W}({\mathcal R}_c)\) is a complete lattice where \({\mathcal L}(R\times S)\) and \({\mathcal L}(R\otimes S)\) are the least upper bound (lub) and the greatest lower bound (glb), respectively, of \({\mathcal L}(R)\) and \({\mathcal L}(S)\) in \({\mathcal W}({\mathcal R}_c)\), and glb of an infinite family \(\{{\mathcal L}(R_j)\}_{j\in J}\) in \({\mathcal W}({\mathcal R}_c)\) can be formed by finite tensor products of rings in \(\{R_j\}_{j\in J}\). Further the rings \(R\) of characteristic zero are considered and for every prime \(p\) the associated ring \(R_p\) is constructed such that \({\mathcal L}(R_p)\subseteq{\mathcal L}(R)\) and \({\mathcal L}(R)\) is the join in \({\mathcal W}\) of \({\mathcal L}(\mathbb{Q})\) and \({\mathcal L}(R_p)\) for all primes \(p\). The relation \(R\precsim S\) is equivalent to \({\mathcal L}(R_p)\subseteq{\mathcal L}(S_p)\) for all primes \(p\) and \(\text{char}(R)\) divides \(\text{char}(S)\) or \(\text{char}(S)=0\).