Primeness Conditions on the Lattice of Hereditary Pretorsion Classes#

Abstract

A lattice ordered monoid is a structure ⟨L;⊕, 0 L ; ≤⟩ where ⟨L;⊕, 0 L ⟩ is a monoid, ⟨L; ≤ ⟩ is a lattice and the binary operation ⊕ distributes over finite meets. If R is an arbitrary ring with identity, then the set ptors R of all hereditary pretorsion classes in the category of right R-modules is an example of a lattice ordered monoid. It is known that right strongly prime and right strongly semiprime rings are characterizable by means of first order sentences in the language of ptors R. A notion of weak primeness and weak semiprimeness is introduced. These are defined by means of first order sentences in ptors R. It is shown that in the presence of a weak commutativity condition, which is also defined in the language of ptors R, the notions of weak primeness [resp. weak semiprimeness] and strong primeness [resp. strong semiprimeness] coincide. It is also proved that within the class of commutative rings, Quasi–Frobenius rings are characterizable by means of a simple sentence in ptors R.