Duprime and Dusemiprime Modules
نویسندگان
چکیده
A lattice ordered monoid is a structure 〈L;⊕, 0L;≤〉 where 〈L;⊕, 0L〉 is a monoid, 〈L;≤〉 is a lattice and the binary operation ⊕ distributes over finite meets. If M ∈ R-Mod then the set ILM of all hereditary pretorsion classes of σ[M ] is a lattice ordered monoid with binary operation given by α :M β := {N ∈ σ[M ] | there exists A ≤ N such that A ∈ α and N/A ∈ β}, whenever α, β ∈ ILM (the subscript in :M is omitted if σ[M ] = R-Mod). σ[M ] is said to be duprime (resp. dusemiprime) if M ∈ α :M β implies M ∈ α or M ∈ β (resp. M ∈ α :M α implies M ∈ α), for any α, β ∈ ILM . 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 ILM , such as coatomicity and compactness.
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