The Lattice Structure of Preradicals Ii: Partitions

Abstract

We continue the study of the preradicals of a ring in the lattice point of view. We introduce several interesting preradicals associated to a given preradical and some partitions of the whole lattice in terms of preradicals. As an application, we also give some classification theorems.

Some operators and dimensions in modular meet-continuous lattices

“…For an inflator j on an idiom A, we say that j is a nucleus if it is an idempotent pre-nucleus. By induction and idiom distributivity it follows that if A is an idiom, then: (8) If f is a pre-nucleus on A, then f α is a pre-nucleus for all ordinal α; in particular f ∞ is a nucleus.…”

“…In [8] the authors introduce four operators in the (big) lattice R-pr. They associate two operators to the product of preradicals τ · σ, and two operators to the coproduct of preradicals (τ : σ).…”

“…The product of preradicals τ and σ is just the composition (στ )(M ) = σ(τ (M )) and the coproduct is the preradical (σ : τ ) given by the submodule (σ : τ )(M ) such that (σ : τ )(M )/σ(M ) = τ (M/σ(M )) with M a module. The authors in [8] introduce for a preradical σ two prearadicals associated to each one of these two operation respectively, these preradicals are the equalizer of σ, the annihilator of σ (these two associated to the product); the co-equalizer of σ and the totalizer of σ (these associated to the coproduct). In particular for the coproduct the following comparison holds σ ∨ τ ≤ (σ : τ ) as in [8] and [9] this new preradicals are useful for describing certain kind of intervals of preradicals and characterize left exact preradicals in terms of some classical elements in a lattice for example pseudocomplements.…”

“…The authors in [8] introduce for a preradical σ two prearadicals associated to each one of these two operation respectively, these preradicals are the equalizer of σ, the annihilator of σ (these two associated to the product); the co-equalizer of σ and the totalizer of σ (these associated to the coproduct). In particular for the coproduct the following comparison holds σ ∨ τ ≤ (σ : τ ) as in [8] and [9] this new preradicals are useful for describing certain kind of intervals of preradicals and characterize left exact preradicals in terms of some classical elements in a lattice for example pseudocomplements. Now starting with a complete modular meet-continuous lattice A consider a monotone function d : A → A such that a ≤ d(a) for all a ∈ A call this function an inflator and let I(A) be the set of all inflators on A. I(A) is a complete lattice in the usual way and I(A) have an extra operation namely the product of inflators this product is the composition.…”

Finite Lattices of Preradicals and Finite Representation Type Rings

“…The unadorned term R-module means left R-module. The terminology and basic concepts about preradicals can be found in [12] and [13]. A (left) preradical over the ring R is a subfunctor of the identity functor on R-Mod.…”