ON LATTICES ASSOCIATED TO RINGS WITH RESPECT TO A PRERADICAL

. We introduce some new lattices of classes of modules with respect to appropriate preradicals. We introduce some concepts associated with these lattices, such as the σ -semiartinian rings, the σ -retractable modules, the σ - V -rings, the σ -max rings. We continue to study σ -torsion theories, σ -open classes, σ -stable classes. We prove some theorems that extend some known results. Our results fall into well known situations when the preradical σ is chosen as the identity preradical.


Introduction
Lattices and big lattices of module classes has been studied to obtain information about the underlying ring R and about its associated module category.For example, the big lattice of preradicals and some associated lattices of special kinds of preradicals have provided a wealth of information about the rings and their module categories.
Similar considerations can be made about module classes lattices defined by closure properties.Some examples of these lattices are: the lattice of the natural classes, the lattice of the hereditary torsion classes, the lattice of Serre classes, that of the Wisbauer classes and some others.
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.
Our objective in this work is to introduce some new lattices of module classes with respect to a preradical σ, to use these lattices to set properties for rings and for their module categories.We introduce the σ-semiartinian rings, the σ-retractable modules, σ-V -rings, and σ-max rings.We extend some well known results in the literature.
A preradical on R-mod is an assignment σ : R-Mod → R-Mod such that for each σ is a subfunctor of the identity functor on R-Mod.R-pr denotes the collection of all preradicals on R-Mod.
Let us recall that σ is a radical if and only if σ(M/σ(M )) = 0, for each M ∈ R-Mod.σ is t-radical if and only if σ(M ) = σ(R)M .t-radicals are precisely the preradicals preserving epimorphisms.t-radicals are also called cohereditary radicals.
A preradical σ is a left exact preradical if it is a left exact functor.This is equivalent to the condition that for each submodule N of a module M we have that σ(N ) = N ∩ σ(M ).That σ is a left exact preradical it is also equivalent to that σ is an idempotent preradical and T σ is a hereditary class.
We will denote R-id, R-rad, R-lep, R-radid the collections of idempotent preradicals, of radicals, of left exact preradicals, and of idempotent radicals, respectively.
Each preradical σ has associated the class This class is closed under quotients and direct sums and it is called the σ-pretorsion class.
Let us recall that a class of R-modules is a pretorsion free class if it is closed under taking submodules and direct products.Each σ ∈ R-pr has associated the pretorsion free class We say that a module M splits in a preradical This implies that σ(Q) is an injective (projective) module.We say that σ centrally splits if for each R-module M we have that M = σ(R)M ⊕ M ′ , with , for further information, see [5], Chapter I.
We say that a two sided ideal I of a ring R is pure if IJ = I ∩ J for every ideal J of R. For a two sided ideal I we have that I is a pure ideal ⇔ for each M ∈ R-Mod [14] Chap.I, §11) ⇔ for all a ∈ I, a ∈ Ia.Notice that if I is a pure ideal, then the preradical I • − is exact.
(3) σ is a radical and F σ is closed under quotient modules.
If σ is an exact preradical, then σ is a t-radical and σ(R) is a pure ideal because If I ≤ R is a pure ideal, then I defines an exact preradical σ by σ(M ) = IM .

Classes of modules.
A lattice L is bounded if it has a smallest element (usually denoted by 0) and a largest element (usually denoted by 1).In a lattice L with 0, an element a * is a pseudocomplement of a ∈ L, if a∧a * = 0 and a * is maximal in L with respect to this property.We say that a * is a strong pseudocomplement of a if it is the largest element in L with respect to a ∧ a * = 0.
We will denote Skel(L) = {a * | a ∈ L} and we will call it the skeleton of L. In a bounded lattice L, we will say that a If L is a proper class instead of a set, we will say that L is a big lattice.
A class of left R-modules is called an abstract class if it is closed under taking isomorphic copies of its members.We consider some closure properties of a class of modules, like being closed under submodules, quotients, extensions, direct sums, injective hulls, products or projective covers, we will use the symbols ≤, ↠, ext, ⊕, E, , P respectively, to abbreviate.If A denotes a set of these closure properties, we denote L A the proper class of classes of modules closed under each closure property in A. So, L {≤} denotes the proper class of hereditary classes in R-mod, L {≤,⊕,E} denotes the class of natural classes, and so on.

ERWIN CERDA-LE ÓN AND HUGO RINC ÓN-MEJ ÍA
We should notice that L A becomes a complete big with inclusion of classes as the order and with infima given by intersections.We will denote ξ A (C ) the The big lattice of torsion theories is denoted by R-TORS (see [14], Chapter VI), and the lattice of hereditary torsion theories is denoted by R-tors (see [9]).Often it will be convenient to identify each torsion theory with its torsion class, that is, R-TORS= L {↠,⊕,ext} and R-tors= L {≤,↠,⊕,ext} .We denote R-jtors For a module class a, we denote ξ(a) the least hereditary torsion theory containing a, and by χ(a) the largest hereditary torsion theory such that each one of its modules has no nonzero submodules in a.

σ-(R-tors) and σ-(R-TORS)
Let us take a preradical σ.We will say that a class C ⊆ R-Mod is σ-hereditary (σ-cohereditary) if it has the following two conditions: F σ ⊆ C , and for each M ∈ C and N ≤ M (M ↠ N ) it happens that σ(N ) ∈ C .We denote L {≤σ} (L {↠σ} ) the collection of all hereditary σ-hereditary (σ-cohereditary) classes.L {≤σ} is a bounded pseudocomplemented big lattice, whose least member is F σ and whose largest member is R-Mod, where infima is given by class intersections.If σ is an idempotent preradical then the pseudocomplements are strong and Skel(L {≤σ} ) is a boolean lattice.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.Let σ be a preradical.
Proof.(⊇): It suffices to show that a class C belonging to the left class is cohered- where (⊆): Suppose C is a module class with the following properties: hereditary, cohereditary, closed under extensions and containing F σ .We want to prove that C is σ-hereditary and σ-cohereditary.First, we show that it is σ-hereditary.
If Lemma 3.12.Let σ be a preradical.Then Proof.Let C be a hereditary module class, we want to show that Proof.(⊇): Let us assume C ∈ R-TORS, we are going to show that , and that As σ (M ) , M/σ (M ) both belong to C because σ is a radical and (1) R is a left semiartinian ring.
(2) Each hereditary torsion theory in R-Mod is generated by a family of simple modules.
Remark 3.15.If R-tors = Skel(L {≤,↠} ) (which happens if and only if R is left semiartinian), then for each T ∈ R-tors there exists a ⊆ R-Simp such that T = Furthermore, for each a ⊆ R-Simp we have that T ξ(a) = T χ(R-Simp\a) .
Remark 3.16.For each centrally splitting preradical σ, we have that For each M ∈ T σ there exists an epimorphism R (X) → M for some set X.As For the other inclusion, notice that for each M ∈ σ(R)-Mod there exists an Proposition 3.17.Let σ be a centrally splitting preradical.The assignment σ * : Analogously it can be shown that The following result is a generalization of Theorem 3.13.
Notice that L σ and R σ are order reversing assignments and that When σ = 1 R-Mod (see [14] Chap.VI), we have that It is known that there exists a bijective correspondence between torsion theories and idempotent radicals, then for all C ⊆ R-Mod we have that L(C ) = T τ and R(C ) = F ν for some idempotent radicals τ, ν, respectively.
Let T be the least torsion containing M , then N ∈ T implies that N ̸ ∈ F hence Hom R (M, N ) ̸ = 0. We conclude that M is retractable.
Remark 4.4.Notice that each retractable R-module is σ-retractable, but a non retractable R-module M can be σ-retractable for some σ ∈ R-pr.
As an example, let t ∈ Z-pr denote the torsion functor and take the Z-module This means that σ(N σ(σ(N )) = 0 follows.It follows that M is a σ-retractable module.□ Example 4.9.Let I ≤ R be a two sided pure ideal and let t(α R I ) ∈ R-pr be defined by t(α R I )(M ) = {m ∈ M | Im = 0}, the annihilator of I on M .Notice that there is a natural isomorphism t(α R I ) ∼ = R/I ⊗ R −.Then t(α R I ) is an exact preradical (R/I is flat).Thus, for all M ∈ R-Mod we have that t Proposition 5.2.Let σ be a cohereditary idempotent preradical.There is an assignment ρ ↠σ : L {≤σ,↠σ} → R-(σ-TORS) defined by Let us take M ∈ ρ ↠σ (C ) and g : M ↠ N .We will show σ(N ) ∈ ρ ↠σ (C ).
Further take f : σ(N ) ↠ L and let us suppose that σ(L) ∈ C and σ(L) ̸ = 0. We have that σ(L) = L (see Remark 5.1).We have the following commutative diagram: where φ is an isomorphism.As σ is cohereditary and f is an epimorphism then We are going to show that M ∈ ρ ↠σ (C ).As σ is idempotent and cohereditary, then it is an idempotent radical and F σ is closed under taking extensions.
If h : M ↠ L is an epimorphism with σ(L) ̸ = 0, we are going to show that σ(L) ̸ ∈ C .We have the following commutative diagram: where π denotes the natural epimorphism, and t : M ′′ ↠ L/hf (M ′ ) is an epimorphism.Notice now that hf : Let us first consider the case where we have the exact sequence from which we obtain the exact sequence )) has to be 0. Thus Now consider the case M ′ ∈ F σ and M ′′ ̸ ∈ F σ .As in the previous case, we obtain Using that σ is idempotent, we obtain that σ (L) = σ (hf (M ′ )) = 0.This contradicts the hypothesis.
(3) If besides σ is a left exact stable preradical, then we have the assignment In [12] is given an assignment between R-Nat and R-tors.From the preceding proposition, for σ = 1 R-Mod , we have the following assignments.
(4) Furthermore, we have the commutative diagram: Recall that we denote R-jtors the collection of all hereditary jansian torsion theories, i.e., the collection of hereditary torsion classes closed under taking products.
Notice that R is a left perfect ring if and only if every hereditary right torsion class is closed under taking products and it is generated by a family of right simple modules (notice the change of side).Thus, it could happen that R-tors ̸ = R-jtors even if R is left perfect.
Let us assume that there exists a monomorphism L ↣ N with L ∈ C and L ̸ = 0. Then we have that the composition L ↣ N ↣ M is a monomorphism with M ∈ ρ ≤ (C ) and L ∈ C .This implies that L = 0, a contradiction.It follows that Now, take M ∈ ρ ≤ (C ) and g : M ↠ N .We are going to show that N ∈ ρ ≤ (C ).
We are going to show that M ∈ ρ ≤ (C ).Let us take a monomorphism L ↣ M with Hence there exists a monomorphism t : L ↣ M ′′ which implies that L = 0, a contradiction.We conclude that L ∈ C implies that L = 0, thus M ∈ ρ ≤ (C ).
Then we have that σ(L) ∈ C implies that σ(L) = 0. We conclude that σ(N ) ∈ Let h : L ↣ M be a monomorphism with σ(L) ̸ = 0. We have the following We have that σ(L) 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 As ker(h) + K = σ(P (M )), we have that ker(h) = σ(P (M )), which implies that h = 0 and L = 0, a contradiction.Hence σ(L) ∈ C implies that L = σ(L) = 0. We conclude that σ(P (M )) ∈ D. □ R is a left Max ring if and only if every conatural class is closed under direct sums (see [2], Theorem 30).Recall that a ring R is left perfect if each left R-modules has a projective cover.If R is a left perfect ring, then each conatural class in R-Mod is generated by a family of simple R-modules (see Corollary 43 of [1]).
Proof.As L/N uc(g) is σ-retractable and σ(S) = S then there exists a nonzero morphism L/N uc(g) → S, which composed with ḡ provides a nonzero R-morphism f : L → S.
We conclude that M is σ-Max.□ This research was supported by a grant from the PostDoctoral Scholarship Program at the National Autonomous University of Mexico.
least class in L A containing C and by χ A (C ) the largest class in L A contained in C .Thus ξ {≤} (C ) denotes the hereditary closure of C , and ξ {↠} (C ) denotes the homomorphic image closure of C .ξ {≤} (C ) will be denoted also S(C ) and ξ {↠} (C ) will be denoted also H(C ).

Lemma 3 . 3 .
Let σ be a radical and C ∈ L {ext} with C ⫆ F σ , then ← σ (C ) ⫅ C .Proof.Take σ and C as in the statement.If A ∈ ← σ (C ) then we have the exact sequence 0
and M ∈ σ * (D).Hence there exist C ∈ C and D ∈ D such that M = σ(C) and M = σ(D), besides M = σ(C) ∈ C and

( 3 )Definition 6 . 1 .S
⇒ (2) It is clear.□6.σ-V-rings and σ-Max-ringsWe generalize the concept of Max-rings and V-rings.Take σ an idempotent preradical.An R-module M is σ-coatomic if each quotient L of M with σ(L) ̸ = 0 has a simple quotient S with σ(S) = S.Let 0 ̸ = Rx ≤ σ(L) be a cyclic module and let us take an epimorphism g : Rx → S, with a simple quotient S. As σ is left exact, then both of Rx and S are of σ-torsion, i.e., S = σ(S).Notice that we have a commutative diagram G G G G G G L/N uc(g).