Duprime and dusemiprime modules

A lattice ordered monoid is a structure left angle bracketL;circled plus,0L;less-than-or-equals, slantright-pointing angle bracket where left angle bracketL;circled plus,0Lright-pointing angle bracket is a monoid, left angle bracketL;less-than-or-equals, slantright-pointing angle bracket is a lattice and the binary operation circled plus distributes over finite meets. If Mset membership, variantR-Mod then the set image of all hereditary pretorsion classes of σ[M] is a lattice ordered monoid with binary operation given byimageα:Mβ colon, equals {Nset membership, variantσ[M] thereexists Aless-than-or-equals, slantN suchthat Aset membership, variantα and N/Aset membership, variantβ},whenever image (the subscript in :M is omitted if σ[M]=R-Mod). σ[M] is said to be duprime (resp. dusemiprime) if Mset membership, variantα:Mβ implies Mset membership, variantα or Mset membership, variantβ (resp. Mset membership, variantα:Mα implies Mset membership, variantα), for any image. The main results characterize these notions in terms of properties of the subgenerator M. It is shown, for example, that M is duprime (resp. dusemiprime) if M is strongly prime (resp. strongly semiprime). The converse is not true in general, but holds if M is polyform or projective in σ[M]. The notions duprime and dusemiprime are also investigated in conjunction with finiteness conditions on image, such as coatomicity and compactness.