Abstract
Given ringsR with prime power characteristicp k, quasivarieties ℒ(R) of lattices generated by lattices of submodules ofR-modules are studied. An algebra of expressionsd not dependent onR is developed, such that each suchd uniquely determines a two-sides ideald R ofR. The main technical result is that ℒ(R)\( \subseteq\)ℒ(S) makes all implications of the formd s =S⇒ dR=R true, for any such expressiond. The proof makes use of the known equivalence between ℒ(R)\( \subseteq\)ℒ(S) and existence of an exact embedding functorR-Mod →S -Mod. Fork ≥ 2, the ordered setW(p k) of all lattice quasivarieties ℒ(R),R having characteristic p K , is shown to be large and complicated, with ascending and descending chains and antichains having continuously many elements. More precisely,W(p k) has a subset which is order isomorphic to the Boolean algebra of all subsets of a denumerably infinite set. Also, given any prime powerp k,k ≥ 2, a ringR can be constructed so that ℒ(R) and ℒ(R op) for the opposite ringR op are distinct elements ofW(p k).
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Research partially supported by Hungarian National Foundation for Scientific Research grant no. 1903.
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Czédli, G., Hutchinson, G. Submodule lattice quasivarieties and exact embedding functors for rings with prime power characteristic. Algebra Universalis 35, 425–445 (1996). https://doi.org/10.1007/BF01197183
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DOI: https://doi.org/10.1007/BF01197183