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.

Key Words:

• Hereditary pretorsion class
• Lattice ordered monoid
• Duprime
• QF

1991 Mathematics Subject Classification:

• Primary 16S90, 16N60
• Secondary 06F05

Acknowledgments

Notes

^#Communicated by E. Puczyłowski.

^aStrictly speaking, the embedding of Id R in ptors R is order reversing. For this reason if Id R is to be viewed as a substructure of ptors R, and this is the perspective we shall offer, then the order relation on Id R must be taken to be reverse set inclusion.