Duprime and dusemiprime modules (Q5956885)

A lattice-ordered monoid is a structure \((L;\oplus,0_L;\leq)\) where \((L;\oplus,0_L)\) is a monoid, \((L,\leq)\) is a lattice and the binary operation \(\oplus\) distributes over finite meets. If \(M\in R\text{-Mod}\) then the set \(L_M\) of all hereditary pretorsion classes of \(\sigma[M]\) is a lattice-ordered monoid with binary operation given by NEWLINE\[NEWLINE\alpha:_M\beta=\{N\in\sigma[M]\mid\exists A\leq N\;A\in\alpha,\;N/A\in\beta\},NEWLINE\]NEWLINE where \(\alpha,\beta\in L_M\). \(\sigma[M]\) is called duprime (resp. dusemiprime) if \(M\in\alpha:_M\beta\) implies \(M\in\alpha\) or \(M\in\beta\) (resp. \(M\in\alpha:_M\alpha\) implies \(M\in\alpha\)), for any \(\alpha,\beta\in L_M\). The main results of this paper characterize these notions in terms of properties of the subgenerator \(M\). The authors show, 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 \(\sigma[M]\) (Theorem 3.3). The authors also investigate the notions of duprime and dusemiprime in conjunction with finiteness conditions on \(L_M\), such as coatomicity and compactness (Theorems 4.9, 4.13).