Duprime and dusemiprime modules

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 M ∈ R-Mod then the set IL M 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 ∈ IL M (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 ∈ IL M. 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 IL M , such as coatomicity and compactness.

Coprime preradicals and modules

“…, this is equivalent to M being strongly prime; that is, each nonzero submodule of M subgenerates M (see [10,Theorem 3.3]). It follows from Corollary 4.6 that this condition is different from M being coprime in M-pr.…”

“…To illustrate this, consider the rationals Q as Z-module: Q is duprime (see [10]) and coprime (as in Definition 4.2) but is not coprime in the sense of [2].…”

“…On the lattice M-hpr of hereditary preradicals on [M], coprimeness with respect to: was investigated in [10] (where the coprime modules are called duprime). As we shall see these notions in general differ from those derived in M-pr since there are obviously more preradicals than left exact preradicals.…”

“…We learn that the condition for 1 to be coprime in M-pr in general is stronger than to be coprime in the lattice of left exact preradicals (see Remarks 4.7). Applied to the ring R, the first condition forces the ring to be simple (see Theorem 3.10), whereas the latter condition requires R to be a left strongly prime ring (see [10,Theorem 3.3]). …”