Big lattices of module classes induced by preradicals

Abstract

We introduce new big lattices of classes of R-modules induced by suitable preradicals, the big lattice of σ-hereditary classes, the big lattice of σ-cohereditary classes, and the big lattice of σ-open classes. We prove that their Skeletons are boolean lattices which generalize and extend the lattice of natural classes, the lattice of conatural classes and the Skeleton of the lattice of hereditary torsion theories, respectively. We also introduce the lattice of σ-hereditary torsion theories, for a exact preradical σ. In case that σ be the identity preradical, we obtain the usual lattice of hereditary torsion classes.

On lattices associated to rings with respect to a preradical

“…(2) ⇒ (3) Since every left perfect ring is left Max, in this case, every conatural class C is closed under direct sums, this implies that ← σ (C ) is closed under direct sums (see [6], Proposition 4). [6], Proposition 15) we have that all σ-conatural class is closed under direct sums.…”

“…If σ is an idempotent cohereditary preradical, then L {↠σ} is a strongly pseudocomplemented big lattice. The big lattice of σ-open classes is [6]). Definition 3.1.…”

“…In [6], the big lattices of module classes induced by a preradical σ over R-Mod were introduced, for example, the lattices of σ-hereditary classes, of σ-cohereditary classes, of σ-natural classes, and of σ-conatural classes. Note that the σ-open classes lattice and the σ-torsion theories lattice were also introduced in the same paper.…”

“…Then there exists a monomorphism Rl ↣ M α1 with Rm α1 ∈ C (see Proposition 6 in [6]). This implies that R α1 = 0, and thus m α1 = 0, a contradiction to the choice of k. Hence L ∈ C and L = 0.…”

“…Let h : L ↣ M be a monomorphism with σ(L) ̸ = 0. We have the following The lattice R-(σ-Conat) = Skel(L {↠σ} ), for an exact and costable preradical σ, is defined in [6] . Where the strong pseudocomplement of C ∈ L {↠σ} is given by (see [2], Theorem 30).…”