a generalization of quantales with applications to … · martha lizbeth shaid sandoval-miranda...

arX

iv:1

511.

0916

9v3

[mat

h.R

A]

31 O

ct 2

016 A GENERALIZATION OF QUANTALES WITH APPLICATIONS TO

MODULES AND RINGS

MAURICIO GABRIEL MEDINA-B ARCENASLUIS ANGEL ZALDIVAR-CORICHI

MARTHA LIZBETH SHAID SANDOVAL-MIRANDA

ABSTRACT. In this paper we introduce a lattice structure as a generalizationof meet-continuous lattices and quantales. We develop a point-free approach tothese new lattices and apply these results toR-modules. In particular, we give amodule counterpart of the well known result that in a commutative ring the setof semiprime ideals, that is, radical ideals is a frame.

1. INTRODUCTION

In general, for the study of rings and modules, the complete lattices of submod-ules play an important role, there are various forms for the examination of theselattices, like via pure lattice theory or in a categorical point of view. In the studyof rings there are other lattice-structures in play, like preradicals, torsion theories,linear filters etc. In the eye of this doctrine, we introduce aframework of latticestructure theory to analyze the submodules of a given moduleand preradicals ona ring, in particular we specialize in the lattice of submodules and we give somecharacterizations of these.

In [17] and [16], the authors introduce some order-lattice structures forstudyingrings, topological spaces and more. In particular, in [17], the author considers lat-tices equipped with a product in which an absolute distribution holds with respectto take suprema and certain comparisons between the∧ and this product must behold. He calls these kind of latticescarries, an by the definition these ones areparticular examples ofquantales. From this theory one can describe interestingphenomena of ring theory likeradical theory. This is one of our motivations inthe scope of a more general situation, which is determinate radicals in the senseof rings but now for modules. Other important structures forstudying rings andmodules areidiomsand frames. Recall that idioms are complete modular meet-continuous lattices, therefore they are a kind of generalization of frames. Anothergeneralization of frames are quantales. With this in mind, one of our first goals isto introduce a generalization of idioms and quantales in which both of them areparticular cases.

This paper is organized as follows: in section2 we give a brief introduction ofthe notation and some examples that motivate our study, in section 3 we introducea general framework for the study of quantales and idioms, wegeneralize someresults of localic nuclei in quantale theory. In last section, we apply these notionsto the idiom of submodules and we generalize some classical structure theorems incommutative ring theory .

1

http://arxiv.org/abs/1511.09169v3

2 MEDINA, ZALD IVAR, SANDOVAL

2. PRELIMINARIES AND BACKGROUND MATERIAL

Throughout this paper, we will work with complete lattices in the usual sense.Recall that a

∨

-semilattice is an structure(A,≤,∨

, 0), where(A,≤) is a partial or-dered set with bottom element0 and a binary operation∨ such thatx∨y ≤ z if andonly if x ≤ z andy ≤ z, for all, a, y, z ∈ A. Also all the

∨

-semilattices we consid-ered here are complete with respect∨. A morphisms of

∨

-semilattices,f : A → Bis a monotone function that preserves arbitrary suprema, that is,f [

∨

X] =∨

f [X]for all X ⊆ A. Then, by the adjoint functor theorem it follows that any morphismof∨

-semilattices, namelyf : A → B, has a right adjointf∗ : B → A. Denotingf = f∗, the compositiond = f∗f

∗ : A → A is a closure operation, that is,d isa monotone function,a ≤ d(a) anddd = d. It can be shown that any of theseoperators rises naturally in this way.

Here our analysis is concentrated in certain kind of∨

-semilaticces. Those arethe following:

A latticeA is meet-continuousif

(IDL), a ∧(

∨

X)

=∨

{a ∧ x | x ∈ X}

for all a ∈ A andX ⊆ A any directed set. Here,X is directedif it is non-emptyand for eachx, y ∈ X there is somez ∈ X with x ≤ z andy ≤ z.

A latticeA is modular if (a ∨ c) ∧ b = a ∨ (c ∧ b), for all a, b, c ∈ A such thata ≤ b.

An idiom is a meet-continuous modular lattice. An important class ofidiomsare the well knownframes. Recall that a complete latticeA is aframeif

(FDL) a ∧(

∨

X)

=∨

{a ∧ x | x ∈ X}

holds for everya ∈ A and everyX ⊆ A.On the other hand, for any latticeA, an implication is a two placed operation

( ≻ ) given byx ≤ (a ≻ b) ⇔ x ∧ b ≤ a, for all a, b ∈ A.The distributivity law (FDL) characterizes frames as follows.

Proposition 2.1. A complete latticeA is a frame if and only ifA has an implica-tion.

This implies that in a frameA any elementa ∈ A has anegation(a ≻ 0) orsimply ¬a. A proof of 2.1 and general background about idioms and frames canbe viewed in [19], [11] and[20]. Some interesting examples of idioms and framesare:

Example 2.2.(1) Given an associative unital ringR consider the category of leftR-modules,

R−Mod and for everyM ∈ R−Mod, denote byΛ(M) the left submod-ules ofM. It is well known thatΛ(M) is a complete meet-continuousmodular lattice, whence an idiom.

(2) Consider any topological spaceS and denote byO(S) the open subsets ofS. This is the generic non-trivial example of a frame.

So any morphism of idioms (in particular of frames) as subcategories of the cate-gory of

∨

-semilattices gives closure operators, we shall summarizesystematicallythe definition of these operators.

An inflator in a idiom A is a functiond : A → A such thatx ≤ d(x) andx ≤ y ⇒ d(x) ≤ d(y). A pre-nucleusd onA is an inflator such thatd(x ∧ y) =

A GENERALIZATION OF QUANTALES WITH APPLICATIONS TO MODULESAND RINGS 3

d(x)∧ d(y). A stableinflator onA is an inflator such thatd(x)∧ y ≤ d(x∧ y) forall x, y ∈ A. Let us denote byI(A) the set of all inflators onA and letP (A) bethe set of all pre-nuclei andS(A) the set of all stable inflators. Clearly,P (A) ⊆S(A) ⊆ I(A). Note that from the definition of inflator, the composition ofanytwo inflators is again an inflator. In fact,I(A) is a poset with the order given ford, d′ ∈ I(A) by d ≤ d′ ⇔ d(a) ≤ d′(a), for all a ∈ A. The identityd0 of A andthe constant functiond(a) = 1 for all a ∈ A, are inflators. Furthermore, these onesare pre-nuclei.

For any two inflatorsd, d′ onA and for alla ∈ A we havea ≤ d(a). From themonotonicity ofd′ it follows thatd′(a) ≤ d′(d(a)), and again we haved′(d(a)) ≥d(a). Therefore, ifA is an idiom, then for any two inflatorsd, d′ ∈ I(A) we havethatd′ ∨ d ≤ d′d. It follows the next:

Lemma 2.3. If A is an idiom andd, d′, k are inflators overA, then:(1) If d ≤ d′, thenkd ≤ kd′ anddk ≤ d′k.(2) kd′ ∨ kd ≤ k(d′ ∨ d) andk(d′ ∧ d) ≤ kd′ ∧ kd.(3) Moreover, ifD ⊆ I(A) is non empty, then:

(a) (∨

D)k =∨

{dk | d ∈ D},(b) (

∧

D)k =∧

{dk | d ∈ D}.

The properties of the composition of inflators gives a crucial construction:Given an inflatord ∈ I(A), let d0 := d0, dα+1 := d ◦ dα for a non-limit ordinal

α, and letdλ :=∨

{dα | α < λ} for a limit ordinal λ. These are inflators, andfrom the comparisond ∨ d′ ≤ dd′, we have a chain of inflators

d ≤ d2 ≤ d3 ≤ . . . ≤ dα ≤ . . . .

By a cardinality argument, there exists an ordinalγ such thatdα = dγ , for α ≥ γ.In fact, we can chooseγ the least of these ordinals, say∞. Thus,d∞ is an inflatorwhich not only satisfiesd ≤ d∞ but alsod∞d∞ = d∞, that is,d∞ is an idempotentor closure operator overA. Denote byC(A) the set of all closure operators. Thisis a poset in which the infimum of a set of closure operators is again a closureoperator. Hence, it is a complete lattice, and the construction we just made definesan operator( )∞ : I(A) → C(A). It is clear that this operation (in the secondlevel) is inflatory and monotone. The supremum inC(A) for an arbitrary familyof closure operators can be computed as follows: first, take anon-empty subset ofclosure operatorsC onA. By the observations of the supremum,

∨

C is an inflator,so we can apply the construction above and obtain a closure operator(

∨

C)∞, thisis the supremum of the familyC in C(A). For an inflatorj over an idiomA, we saythatj is anucleusif is an idempotent pre-nucleus. By induction and distributivityon idioms, it follows that ifA is an idiom, then:

(1) If f is a pre-nucleus onA, thenfα is a pre-nucleus. In particular,f∞ is anucleus.

(2) If f is a stable inflator, then eachfα is a pre-nucleus, and for limit ordinalsλ, fλ is a pre-nucleus. In particular,f∞ is a nucleus.

One of the most important observations in the inflator theoryis:

Theorem 2.4. For any idiomA, denote byN(A) the set of all nuclei onA then,the complete latticeN(A) is a frame.

Most of the structural analysis of idioms is related with theframeN(A), asranking techniques, dimensions and other point free-theoretical constructions. In


particular, every nucleusj onA determines a quotient of the idiom, the set of allfixed points ofj, Aj together with an idiom morphismj∗ : A → Aj given byj∗(a) = j(a).

There are other∨

-semilattices structures that we need to mention.LetQ be a

∨

-semilattice,Q is aquantaleif Q has a binary associative operation· : Q×Q → Q such that

(LQ) l(

∨

X)

=∨

{lx | x ∈ X} , and

(RQ)(

∨

X)

r =∨

{xr | x ∈ X}

hold for all l, r ∈ Q andX ⊆ Q.Notice that if In the previous definition we assume meets instead of joins, then

we obtain the dual quantale with respect to meets.In case that only (LQ) (respectely (RQ)) holds, some authorsjust say thatQ is

a left quantale (respectely right quantale).Note that any frame is a quantal, with the operation∧, in fact:

Proposition 2.5. Let (Q,≤,∨

, 0, ·) be a quantale. Then,Q is a frame if and onlyif (Q, ·, e) is a monoid with unital elemente = 1 and each element is idempotentwith respect·, here1 is the top of the complete lattice(Q,≤,

∨

, 0, 1).

Now the morphisms of quantales are∨

-morphisms such that respect the prod-uct. As in the case of idioms, we have the concept of inflator. In particular, wehave that an inflatord : Q → Q on a quantale isstable if ld(x) ≤ d(lx) andd(x)r ≤ d(xr) for all x, l, r ∈ Q. A pre-nucleusin Q is an inflatord such thatd(x)d(y) ≤ d(xy) holds for allx, y ∈ Q. In this sense, aquantic nucleuson Qis an idempotent pre-nucleusj on Q. In general, the set of all nucleiN(Q) is acomplete lattice which is not a quantale, but the complete lattice structure ensuresthat there exists a unique quantic nucleusσ ∈ N(Q) such that for each quantic nu-cleusk ∈ N(Q), the set of fixed points ofk, Qk, that is a quotient ofQ, is a frameif and only if σ ≤ k. The latter is proved in [16, Lemma 3.2.5]. There are moreinteresting families of inflators controlling the product and its comparison with∧onQ.

Some examples of quantales, as we mentioned before, are frames. Other exam-ples come from ring theory. The lattice of left ideals, the one of right ideals andthat of two sided ideals with the usual product, are examplesof quantales. On theother hand, notice thatR−fil the lattice of all preradical filters with the operationof Gabriel multiplication of preradical filters satisfies (LQ), i.e. is a left quantale.See [7, 3.13 Proposition].

3. QUASI-QUANTALES

Definition 3.1. Let A be a∨

-semilattice. We say thatA is aquasi-quantaleif ithas an associative productA×A → A such that for all directed subsetsX,Y ⊆ Aanda ∈ A:

(RDQ)(

∨

X)

a =∨

{xa | x ∈ X},

and

(LDQ) a(

∨

Y)

=∨

{ay | y ∈ Y }.


We sayA is a left-unital (resp. right-unital, resp. bilateral-unital) quasi-quantaleif there existse ∈ A such thate(a) = a (resp.(a)e = a, resp.e(a) = a = (a)e)for all a ∈ A.

Example 3.2.(1) If A is an idiom, thenA is a bilateral-unital quasi-quantale with∧ : A ×

A → A.(2) LetR be a ring with identity and letBil(R) = {I ⊆ R | I bilateral ideal}.

ThenBil(R) is a bilateral-unital quasi-quantale with the product of ideals.Notice thatRI = I = IR for all I ∈ Bil(R). Moreover,Bil(R) satisfies

∑

J

Ii

J =∑

J

(IiJ)

and

J

∑

J

Ii

=∑

J

(JIi),

for all {Ii}J ⊆ Bil(R).Now, note thatΛ(R) = {I ⊆ R | I is a left ideal} satisfies the latter

distributive laws but it is just a left-unital quasi-quantale.(3) Let R-pr be the big lattice of preradicals onR−Mod, where the order

is given as follows:r, s ∈ R−pr if and only if r(M) ≤ s(M) for allM ∈ R−Mod. Their infimumr ∧ s and supremumr ∨ s are defined as(r∧ s)(M) = r(M)∩ s(M)) and(r∨ s)(M) = r(M)+ s(M)) for everyM ∈ R−Mod, respectively. In fact, by [12, Theorem 8, 1.(b)] it followsthatR−pr is a bilateral-unital quasi-quantale which is not a quantale with∧ as product.

On the other hand, givenr, s ∈ R−pr consider their productrs whichis defined as(rs)(M) = r(s(M)), for everyM ∈ R−Mod. Notice thatR−pr with this product just satisfies (RQD). See [12, Theorem 8, 2.(b)].

(4) Let A be a lattice. The set of all inflators onA, denoted byD(A) justsatisfies (RQ) so in particular (RDQ). See Lemma2.3(3) (a) and [20].

(5) LetM be a leftR-module. GivenN,L ∈ Λ(M), in [3] was defined theproduct

NML =∑

{f(N)|f ∈ HomR(M,L)}.

In general, this product is not associative but ifM is projective inσ[M ]then it is. Therefore, ifM is projective inσ[M ], thenΛ(M) is a quasi-quantale. In fact, for all subset{Ni}J,

(∑

J

Ni)ML =∑

J

(NiML).

Let Λfi(M) = {N ≤ M | N is fully invariant}, that is,f(N) ⊆ N forany endomorphismf of M . Notice thatΛfi(M) is a right-unital quasi-quantale, becauseNMM = N for all N ∈ Λfi(M). This example will bedetailed in section4.

(6) Any quantale is a quasi-quantale.


(7) The lattice of all preradical filters, denoted byR−fil with the operation ofGabriel multiplication of preradical filters satisfies (LQ)and (RDQ), re-spectively. Therefore,R− fil, is a quasi-quantale which is not a quantale.See [7, 3.13 Proposition].

Proposition 3.3. LetA be a quasi-quantale andx, y, z ∈ A. Then, the followingconditions hold.

(1) If x ≤ y, thenzx ≤ zy andxz ≤ yz.(2) Moreover, if1a ≤ a for all a ∈ A, then

(a) xy ≤ y ∧ x1(b) x0 = 0

Proof. (1) Notice that{x, y} ⊆ A is a directed subset, sozy = z(x∨y) = zx∨zy.Thuszx ≤ zy. Analogously,yz = (x ∨ y)z = xz ∨ yz. Soxz ≤ yz.(2) (a) We have that{y, 1} is directed, sox1 = x(y∨1) = xy∨x1. Thenxy ≤ x1.On the other hand,y ≥ 1y = (x ∨ 1)y = xy ∨ 1y = xy ∨ y, thenxy ≤ y. Thusxy ≤ x1 ∧ y.(2)(b) By (1), it follows thatx0 ≤ x1 ∧ 0 = 0. �

Proposition 3.4. LetA be a quasi-quantale. The following conditions hold.(1) If x ≤ y andz ≤ v, thenxz ≤ yv.(2) xy ∨ xz ≤ x(y ∨ z) andyx ∨ zx ≤ (y ∨ z)x, for all x, y, z ∈ A.(3) If 1a ≤ a for all a ∈ A, then

(a) xx := x2 ≤ x1 andx2 ≤ x, for everyx ∈ A.(b) (x ∧ y)2 ≤ xy and(x1 ∧ y)2 ≤ xy.

Proof. (1) It follows from Proposition3.3(2) (a).(2) Sincey ≤ y ∨ z, thenxy ≤ x(x ∨ z). Similarly, z ≤ y ∨ z implies thatxz ≤ x(y ∨ z). Therefore,xy ∨ xz ≤ x(y ∨ z).(3) (a) It follows from Proposition3.3(2) (a).(3) (b) Sincex ∧ y ≤ x andx ∧ y ≤ y, by (2), we conclude that(x ∧ y)2 ≤ xy.On the other hand, sincex1 ∧ y ≤ x1 andx1 ∧ y ≤ y. By hypothesis,(1) andthe associativity of the product, we conclude that(x1 ∧ y)2 ≤ (x1)y = x(1y) ≤xy. �

Proposition 3.5. LetA be quasi-quantale. Consider the following statements.(1) xy = 0 if and only ifx1 ∧ y = 0, for everyx, y ∈ A.(2) If x2 = 0, thenx = 0, for everyx ∈ A.

Then, the condition (1) always implies (2). If in addition,A is a quasi-quantalewhich satisfies that1a ≤ a for all a ∈ A, the two conditions are equivalent.

Proof. First, we prove that (1) implies (2). Letx ∈ A be such thatx2 = 0. Then,by (1) it follows thatx = x ∧ x = 0.

Now, suppose that1a ≤ a for all a ∈ A. It remains to prove that (2) implies (1).Let x, y ∈ A be such thatxy = 0. Then, by Proposition3.4 (3) (b), we have that(x1 ∧ y)2 ≤ xy = 0, so(x1 ∧ y)2 = 0. Hence,x1 ∧ y = 0. Conversely, supposethatx1 ∧ y = 0. Then, it is immediately thatxy ≤ x1 ∧ y = 0. �

Recall that a latticeA is pseudomultiplicativeif it has a binary product(x, y) 7→xy which satisfies the following conditions forx, y, z ∈ A:

(1) If x ≤ y thenxz ≤ yz andzx ≤ zy.


(2) xy ≤ x ∧ y.(3) x(y ∨ z) ≤ xy ∨ xz.

See [9, Section 4] for more details.From Proposition3.3, it follows that a left-unital quasi-quantale (withe = 1)

generalizes the concept of a pseudomultiplicative lattice. Even though letA be abilateral quasi-quantale withe = 1, this one just will satisfy the above conditions(1) and (2). Indeed, a quasi-quantale, in general, does not hold the latter condition(3), see Proposition3.4(3) (b).

Lemma 3.6. LetA be a quasi-quantale which satisfies the following identities:

a(b ∨ c) = ab ∨ ac

(b ∨ c)a = ba ∨ ca

for all a, b, c ∈ A. ThenA is a quantale.

Proof. Let X ⊆ A. DefineY = {x1 ∨ ... ∨ xn | xi ∈ X}. ThenX ⊆ Y , so∨

X ≤∨

Y andY is a direct subset ofA. SinceA is quasi-quantale

a(

∨

Y)

=∨

{ay | y ∈ Y }.

Everyy ∈ Y is of the formy = x1∨...∨xn with xi ∈ X. Sincex1∨...∨xn ≤∨

Xthen

∨

{ay | y ∈ Y } ≤ a(

∨

X)

≤ a(

∨

Y)

,

soa(

∨

X)

=∨

{ay | y ∈ Y }.

We have thatX ⊆ Y , whence∨

{ax | x ∈ X} ≤∨

{ay | y ∈ Y }. On the otherhand, by hypothesis

ay = a(x1 ∨ ... ∨ xn) = ax1 ∨ ... ∨ axn ≤∨

{ax | x ∈ X},

thus∨

{ay | y ∈ Y } ≤∨

{ax | x ∈ X}. Hence

a(

∨

X)

=∨

{ax | x ∈ X}

�

Lemma 3.7. Let (A,≤,∨, 1) be a complete lattice. Then,A is meet-continuous ifand only ifA is a quasi-quantale such that(a) The binary operation inA is commutative.(b) A has an identity ,e = 1

Proof. First, suppose thatA is a meet-continuous complete lattice. It is clear thatA with the binary operation∧ is a quasi-quantale which satisfies (a) and (b).

Conversely, sinceA is commutative, it follows that1a = a = a1. This impliesthatab ≤ a ∧ b. Thusab = a ∧ b. Therefore,A is meet-continuous. �

Definition 3.8. Let A be a quasi-quantale. Considers : A → A a inflator onA.We say that:

(1) s is contextual stableif s(a)x ≤ s(ax) and y(s(a)) ≤ s(ya) for alla, x, y ∈ A.

(2) s is pre-multiplicativeif s(a) ∧ b ≤ s(ab) for all a, b ∈ A.(3) s is multiplicative if s(a) ∧ s(b) = s(ab) for all a, b ∈ A.


(4) s is acontextual pre-nucleusif s(a)s(b) ≤ s(ab) for all a, b ∈ A.(5) s is acontextual nucleusif s2 = s ands is a contextual pre-nucleus.

Let A be a quasi-quantale andj : A → A a nucleus onA, we define a binaryoperation inAj given bya, b ∈ Aj , a · b = j(ab).

Proposition 3.9. LetA be a quasi-quantale andj be a contextual nucleus. Then(Aj , ·) is a quasi-quantale. IfA is a left-unital quasi-quantale with identitye thenAj is a left-unital quasi-quantale with identityj(e).

Proof. Let a, b, c ∈ Aj. Consider the following inequalities,

(1) abc ≤ j(ab)c ≤ j(j(ab)c) = j(ab) · c = (a · b) · c

(2) j(ab)c = j(ab)j(c) ≤ j(abc)

Using (2), we have

(3) (a · b) · c = j(j(ab)c) ≤ j2(abc) = j(abc).

By (1), j(abc) ≤ (a · b) · c, and by(3), we have the equality

(a · b) · c ≤ j(abc) ≤ (a · b) · c

Analogously,a · (b · c) = j(abc). Thena · (b · c) = (a · b) · c.

Now, letX ⊆ Aj be a directed subset anda ∈ Aj . Then,

a ·

(

j∨

X

)

= j

(

a

j∨

X

)

= j(

aj(

∨

X))

≥ a(

∨

X)

=∨

{ax|x ∈ X}.

Notice thatj(ax) ≤ j(∨

{ax | x ∈ X}) for all x ∈ X, so

j

(

a ·

(

j∨

X

))

≥ j(

∨

{ax|x ∈ X})

≥∨

{j(ax)|x ∈ X}.

Sincej is a nucleus,

a·

(

j∨

X

)

= j

(

a ·

(

j∨

X

))

≥ j(

∨

{j(ax) | x ∈ X})

=

j∨

{a·x | x ∈ X}.

On the other hand,j∨

{a · x | x ∈ X} = j(

∨

{j(ax) | x ∈ X})

≥ j(

∨

{ax | x ∈ X})

= j(

a∨

X)

≥ j(a)j(

∨

X)

= a ·

j∨

X

Thus

a ·

j∨

X ≥

j∨

{ax|x ∈ X} ≥ a ·

j∨

X

Hence(Aj , ·) is a quasi-quantale.Now, suppose thatA is a left-unital quasi-quantale. Leta ∈ Aj, then

a = ea ≤ j(e)a ≤ j(j(e)a) = j(j(e)j(a)) ≤ j(ea) = j(a) = a.

Soa = j(j(e)a) = j(e) · a. HenceAj is a left-unital quasi-quantale. �


Proposition 3.10.LetA be a quasi-quantale. Then, for each multiplicative nucleusd onA, the setAd is a meet-continuous lattice.

Proof. LetX ⊆ Ad be a directed subset anda ∈ Ad. Then,

d((

∨

X)

∧ a)

= d((

∨

X)

a)

= d(

∨

{xa|x ∈ X})

≤ d(

∨

{d(xa)|x ∈ X})

= d(

∨

{d(x) ∧ d(a)|x ∈ X})

= d(

∨

{x ∧ a|x ∈ X})

≤ d((

∨

X)

∧ a)

.

Therefore,d((∨

X) ∧ a) = d (∨

{x ∧ a|x ∈ X}). Thus(

d∨

X

)

∧ a = d(

∨

X)

∧ d(a) = d((

∨

X)

∧ a)

= d(

∨

{x ∧ a|x ∈ X})

=d∨

{x ∧ a|x ∈ X}.

�

Corollary 3.11. Let A be a quasi-quantale. Suppose that for anyX ⊆ A anda ∈ A is satisfied

(

∨

X)

a =∨

{xa | x ∈ X}.

Then, for each multiplicative nucleusd onA, the setAd is a frame.

Remark 3.12. In Corollary3.11, if A is an idiom thenAj is an idiom.

Now, considerp : A → A an inflator onA such that

p(a)p(b) ≤ p(ab) = p(a ∧ b) = p(a) ∧ p(b).

We call suchp, idiomatic pre-nucleus, and denote byIP (A) the set of idiomaticpre-nucleus.

Let S ⊆ IP (A) be a directed subset. Notice that

(∨

S)(a)(∨

S)(b) = (∨

{s(a) | s ∈ S})(∨

{s(b) | s ∈ S})

=∨

{s(a)s′(b) | s, s′ ∈ S}.

SinceS is directed, fors, s′ ∈ S there existsf ∈ S such thats, s′ ≤ f . Then∨

{s(a)s′(b) | s, s′ ∈ S} ≤∨

{f(a)f(b) | f ∈ S}

≤∨

{f(ab) | f ∈ S} =∨

{f(a ∧ b) | f ∈ S}.

Hence,(∨

S)(a)(∨

S)(b) ≤ (∨

S)(ab) = (∨

S)(a ∧ b). Since the supremum ofa directed set of pre-nucleus is a pre-nucleus, we obtain

∨

S ∈ IP (A).Denote byNI(A), the set of idiomatic nucleus.

Proposition 3.13. LetA be a quasi-quantale. ThenNI(A) is a frame.

10 MEDINA, ZALDIVAR, SANDOVAL

Proof. Let j ∈ NI(A) andk a inflator. LetG = {f | f ∧ k ≤ j}. Let f, g ∈ Gandp = fg. For allx ∈ A we have that

k(x) ∧ p(x) = k(x) ∧ f(g(x)) ≤ f(k(x)) ∧ f(g(x))

= f(k(x) ∧ g(x)) ≤ f(j(x)).

Thenk(x) ∧ p(x) ≤ f(j(x)) ∧ k(x) ≤ f(j(x)) ∧ k(j(x))

≤ (f ∧ k)(j(x)) ≤ j(j(x)) = j(x).

Thus,G is directed and whence∨

G is idiomatic.Let h =

∨

G. Consider

j ◦ (h ∧ k)(x) = j(h(x) ∧ k(x)) = j(h(x)k(x)) = j((

∨

G)

(x)k(x))

= j(

∨

{g(x)k(x) | g ∈ G})

≤ j(

∨

{j(g(x)k(x)) | g ∈ G})

≤ j(

∨

{j(g(x) ∧ k(x)) | g ∈ G})

≤ j(j(j(x))) = j(x)

Thus,j ◦ (h ∧ k) = j. Thenh ∧ k ≤ j, that is,h ∈ G. This implies thath2 = h,and soh ∈ NI(A).

Hence,NI(A) has implication. Consequently,NI(A) is a frame. �

Definition 3.14. Let A be a quasi-quantale. An element1 6= p ∈ A is prime ifwheneverab ≤ p thena ≤ p or b ≤ p.

Definition 3.15. LetA be a quasi-quantale andB a sub∨

-semilattice. We say thatB is asubquasi-quantale ofA if

(

∨

X)

a =∨

{xa | x ∈ X}

and

a(

∨

Y)

=∨

{ay | y ∈ Y },

for all directed subsetsX,Y ⊆ B anda ∈ B.

Definition 3.16. Let A be a quasi-quantale andB a subquasi-quantale ofA. Anelement1 6= p ∈ A is a prime element relative toB if wheneverab ≤ p witha, b ∈ B thena ≤ p or b ≤ p.

It is clear that a prime element inA is a prime element relative toA.We define the spectrum relative toB of A as

SpecB(A) = {p ∈ A | p is prime relative to B}

If B = A, the we just writeSpec(A).

Lemma 3.17. LetB be a subquasi-quantale of a quasi-quantaleA. Suppose that0, 1 ∈ B and1b, b1 ≤ b for all b ∈ B, then

(1) ab ≤ a ∧ b for all a, b ∈ B.(2) Letp ∈ A a prime element relative toB. If a, b ∈ B, thenab ≤ p if and

only if a ≤ p or b ≤ p.


Proof. (1). It follows by Proposition3.3.(2).(a) thatab ≤ a ∧ 1b. By hypothesis1b ≤ b, soab ≤ a ∧ b.

(2). Let a, b ∈ B andp ∈ A a prime element relative toB. First, if ab ≤ p,then, by definition, it follows thata ≤ p or b ≤ p.

Conversely, suppose thata ≤ p. By (1), ab ≤ a ∧ b ≤ a, then ab ≤ p.Analogously ifb ≤ p. �

From now on, letB be a subquasi-quantale of a quantaleA and suppose that0, 1 ∈ B and1b, b1 ≤ b for all b ∈ B.

Proposition 3.18. Let B be a subquasi-quantale of a quasi-quantaleA. ThenSpecB(A) is a topological space, where the closed subsets are subsetsgiven by

V(b) = {p ∈ SpecB(A) | b ≤ p},

with b ∈ B.In dual form, the open subsets are of the form

U(b) = {p ∈ SpecB(A) | b � p},

with b ∈ B.

Proof. It clear thatU(0) = ∅ andU(1) = SpecB(A). Let {bi}I be a family ofelements inB. Then

U

(

∨

I

bi

)

=

{

p ∈ SpecB(A) |∨

I

bi � p

}

=⋃

I

{p ∈ SpecB(A) | bi � p}

=⋃

I

U(bi)

Now, leta, b ∈ B. Then, by Lemma3.17.(2),

U(ab) = {p ∈ SpecB(A) | ab � p} = {p ∈ SpecB(A) | a � p and b � p}

= ({p ∈ SpecB(A) | a � p}) ∩ ({p′ ∈ SpecB(A) | b � p′}) = U(a) ∩ U(b).

�

Remark 3.19. LetO(SpecB(A)) be the frame of open subsets ofSpecB(A). Wehave an adjunction of

∨

-morphisms

BU

..O(SpecB(A))

U∗

mm

WhereU∗ is defined as

U∗(W ) =∨

{b ∈ B | U(b) ⊆ W}

The compositionµ = U∗ ◦ U is a closure operator inB.

Proposition 3.20. Let b ∈ B. Thenµ(b) is the largest element inB such thatµ(b) ≤

∧

{p ∈ SpecB(A) | p ∈ V(b)}.


Proof. By definition,

µ(b) = U∗(U(b)) =∨

{c ∈ B | U(c) ⊆ U(b)}

=∨

{c ∈ B | V(b) ⊆ V(c)} ≤∧

{p ∈ SpecB(A) | p ∈ V(b)}.

Let x ∈ B such thatx ≤ p for all p ∈ V(b), thenV(b) ⊆ V(x). Thus,x ≤∨

{c ∈A | V(b) ⊆ V(c)}, whencex ≤ µ(b). �

Theorem 3.21.The closure operator inµ : B → B is a multiplicative pre-nucleus.

Proof. Let a, b ∈ B. By Lemma3.17.(1), ab ≤ a ∧ b. Thusµ(ab) ≤ µ(a) ∧ µ(b).By Proposition3.20,

µ(a)∧µ(b)≤(

∧

{q ∈ SpecB(A)|q∈V(a)})

∧(

∧

{q′ ∈ SpecB(A) | q′∈V(b)}

)

.

Let p ∈ SpecB(A) such thatab ≤ p, thena ≤ p or b ≤ p. If a ≤ p then,∧

{q ∈ SpecB(A) | q ∈ V(a)}} ≤ p.

Thus,µ(a) ∧ µ(b) ≤ p. Analogously, ifb ≤ p thenµ(a) ∧ µ(b) ≤ p. Hence

µ(a) ∧ µ(b) ≤∧

{p ∈ SpecB(A) | p ∈ V(ab)}

By Proposition3.20, µ(ab) is the largest element inB less or equal than∧

{p ∈SpecB(A) | p ∈ V(ab)}, thereforeµ(a)∧ µ(b) ≤ µ(ab). Thusµ is multiplicative.

Now, sinceab ≤ a ∧ b thenµ(a) ∧ µ(b) = µ(ab) ≤ µ(a ∧ b). The otherinequality always holds. Thusµ is a pre-nucleus. �

Corollary 3.22. Aµ is an meet-continuous lattice.

Proof. It follows by Proposition3.10. �

4. THE LARGE SPECTRUM

In this section, we introduce a new spectrum for a module and we give somecharacterizations of modules with this space.

Firstly, recall that for any leftR-moduleM, in [3, Lemma 2.1] was defined theproduct of submodulesN,L ∈ Λ(M),

NML :=∑

{f(N)|f ∈ HomR(M,L)}.

Remark 4.1. [4, Proposition 1.3] This product satisfies the following propertiesfor every submodulesK,K ′ ∈ Λ(M):(1) If K ⊆ K ′ thenKMX ⊆ K ′

MX for every moduleX.(2) If X is a left module andY ⊆ X thenKMY ⊆ KMX.(3) MMX ⊆ X for every moduleX.(4) KMX = 0 if and only if f(K) = 0 for all f ∈ Hom(M,X).(5) 0MX = 0, for every moduleX.(6) Let {Xi | i ∈ I} be a family of submodules ofM then

∑

i∈I (KMXi) ⊆

KM

(∑

i∈I Xi

)

.(7)

(∑

i∈I Ki

)

MN =

∑

i∈I KiMN for every family of submodules{Ki | i ∈ I}of M .


Proof. The proof of this is in [4, Proposition 1.3]. Here we only give the proof of(6) and(7).

(6) Let {Xi | i ∈ I} be a family of submodules ofM . SinceXi ≤∑

i∈I Xi,by (2) of this proposition(KMXi) ⊆ KM

(∑

i∈I Xi

)

. Thus,

∑

i∈I

(KMXi) ⊆ KM

(

∑

i∈I

Xi

)

.

(7) Let {Ki | i ∈ I} be a family the submodules ofM . Then(

∑

i∈I

Ki

)

M

N =∑

{(

∑

i∈I

Ki

)

|f ∈ HomR(M,N)

}

=∑

{

∑

i∈I

f (Ki) |f ∈ HomR(M,N)

}

=∑

i∈I

∑

{f(Ki)|f ∈ HomR(M,N)}

=∑

i∈I

(KiMN) .

�

Now, recall that anR−moduleN is said to be subgenerated byM if N isisomorphic to a submodule of anM−generated module. It is denoted byσ[M ]the full subcategory ofR−Mod whose objects are allR−modules subgeneratedby M. Also, a moduleN is called a subgenerator inσ[M ] if σ[M ] = σ[N ]. Inparticular,M is a subgenerator inR−Mod if σ[M ] = R−Mod. See [23, §15] formore details.

Remember thatN ≤ M is a fully invariant submodule ofM if f(N) ≤ N forall f endomorphism ofM . The set of fully invariant submodules ofM is denotedbyΛfi(M).

Remark 4.2. LetM ∈ R−Mod and projective inσ[M ]. The following conditionshold.(1) The product−M− : Λ(M)× Λ(M) → Λ(M) is associative.(2) KM

(∑

i∈I Xi

)

=∑

i (KMXi) for every directed family of submodules{Xi |i ∈ I} of M .

(3) If N,L ∈ Λfi(M), thenNML ∈ Λfi(M) and hence the product−M− is wellrestricted inΛfi(M).

Proof. (1) It follows by [2, Proposition 5.6].(2) Let{Xi|i ∈ I} be a directed family of submodules ofM . Let

∑

j∈J fj(kj) ∈

KM

(∑

i∈I Xi

)

, with fj : M →∑

i∈I Xi. SinceM is projective inσ[M ] for eachfj there existsgij : M → Xi such that

∑

i∈I gij (kj) = fj(kj).

M

fj��

⊕gij

xxqqqqqqqqqqq

⊕

i∈I Xi// //∑

i∈I Xi


Then∑

j∈J

fj(kj) =∑

j∈J

∑

i∈I

gij (kj) ∈∑

Xij .

Since this sum is finite and{Xi|i ∈ I} is directed, there existsl ∈ I such that∑

Xij ⊆ Xl. Thus∑

j∈J

fj(kj) =∑

j∈J

∑

i∈I

gij (kj) ∈ KMXl ⊆∑

i∈I

(KMXi)

The other contention follows by4.1.(6).(3) LetN,L ∈ Λfi(M) andg : M → M . Then

g(NML) = g(

∑

{f(N)|f ∈ HomR(M,L)})

=∑

{gf(N)|f ∈ HomR(M,L)}

SinceL ∈ Λfi(M), gf ∈ HomR(M,L). Hence

g(NML) ⊆ NML.

�

Notice that in Remark4.2(1), the projectivity condition forM is necessary as isshown in the following example which was taken from [3, Lemma 2.1 (vi)]: con-siderZ the additive group of integers andQ the additive group of rational numbers,then0 = (Z Q Z )QQ 6= ZQ (Z QQ) = Q.

Proposition 4.3. Let M ∈ R−Mod. If M is projective inσ[M ], the followingconditions hold.(1) Λ(M) is a quasi-quantale. In fact, for all subset{Ni}J, (

∑

JNi)ML =∑

J (NiML).

(2) Λfi(M) is a right-unital subquasi-quantale ofΛ(M).

Proof. (1) By Remark4.2 (1), the product−M− : Λ(M) × Λ(M) → Λ(M) isassociative, and by Remark4.1(7) and Remark4.2(2),Λ(M) is a quasi-quantale.

(2) Since the product−M− is well restricted inΛfi(M), thenΛfi(M) is asubquasi-quantale. Now, ifN ∈ Λfi(M) then

NMM =∑

{f(N)|f ∈ HomR(M,M)} ⊆ N,

butN = IdM (N) ≤ NMM . HenceNMM = N . ThusΛfi(M) is a right-unitalquasi-quantale. �

In this study, we also have preradicals theory. Recall that apreradical r onR−Mod is a subfunctor of the identity functor onR−Mod, i.e. r assigns to eachM a submoduler(M) in such a way that everyf ∈ HomR(M,N) restricts toa homomorphismr(f) ∈ HomR(r(M), r(N)). In particular,r(R) is a two-sidedideal. The big lattice of preradicals inR−Mod is denoted byR−pr. In fact,R−pris a left -unital quasi-quantale with the product∧, as was mentioned in Example3.2(3).

Let M ∈ R−Mod. For eachN ∈ Λfi(M), there are two distinguished prerad-icals,αM

N andωMN , which are defined as follow

αMN (L):=

∑

{f(N) | f ∈ HomR(M,L)}


andωMN (L):=

⋂

{f−1(N) | f ∈ HomR(L,M)},

for eachL ∈ R−Mod.

Remark 4.4. [12, Proposition 5]. IfN is a fully invariant submodule ofM, thenthe following conditions are satisfied.

(1) The preradicalsαMN andωM

N have the property thatαMN (M) = N andωM

N (M) =N respectively.

(2) The class{r ∈ R−pr | r(M) = N} is precisely the interval[αMN , ωM

N ].

The reader can find more properties of these preradicals in [12], [13], [14] and[15, Proposition 1.3].

Definition 4.5. LetM, P ∈ R−Mod such thatP ≤ M. Define

ηMP : R−Mod → R−Mod

ηMP (L) :=⋂

{f−1(P ) ∈ R−Mod|f ∈ HomR(L,M)},

for eachL ∈ R−Mod.

It is clear thatηMP is a preradical. Also, it is clear thatηMP (M) ≤ P. WhenP ∈ Λfi(M), it follows thatP = ηMP (M). And in this case,ωM

P = ηMP .

Remark 4.6. (1) ηMP � ωMηMP

(M)

(2) If N,L ∈ Λfi(M) andNML ≤ P, thenωM(NML) � ηMP .

Remark 4.7. LetM ∈ R−Mod.

(1) If P,Q ≤ M satisfy thatP ≤ Q, thenηMP � ηMQ .

(2) If r ∈ R−pr andP ≤ M satisfy thatr(M) ≤ ηMP (M), then

r � ωMr(M) � ηMP � ωM

ηMP

(M).

Remark 4.8. Let M be anR-module. We can considerSpec(Λfi(M)). Noticethat the prime submodules defined in [13] are the elements inSpec(Λfi(M)).

Proposition 4.9. Let M ∈ R−Mod andP ≤ M. If ηMP is a prime preradicali.e. ηMP ∈ Spec(R−pr) and ηMP (M) 6= M, then ηMP (M) ∈ Spec(Λfi(M)).Moreover,ηMP (M) is the largest element inSpec(Λfi(M)) which is contained inP.

Proof. LetN,L ∈ Λfi(M) such thatNML ≤ ηMP (M). This is,(αMN ·αM

L )(M) ≤ηMP (M). By Remark4.7(1) it follows thatαM

N · αML � ηMP . SinceηMP is a prime

preradical, we getαMN � ηMP or αM

L � ηMP . Thus,N ≤ ηMP (M) or L ≤ ηMP (M).

Therefore,ηMP (M) ∈ Spec(Λfi(M)).Finally, letK ∈ Λfi(M) be a prime submodule ofM such thatK ≤ P. Then,

ωMK � ηMP . Thus, applyingM in the last inequality we obtainK = ωM

K (M) ≤ηMP (M) ≤ P. �


From now on,M will be assumed projective enσ[M ], in order to have thatΛfi(M) is a right-unital subquasi-quantale ofΛ(M).

Let M be anR-module. IfP is a prime element ofΛ(M) relative toΛfi(M),then we will call it a large prime submodule ofM . We will write LgSpec(M)instead ofSpecΛfi(M)(Λ(M)).

Proposition 4.10. LetM, P ∈ R−Mod such thatP ≤ M. Then, the followingconditions are equivalent.

(1) P is a large prime submodule ofM.(2) ηMP is a prime preradical.

Proof. (1)⇒(2) From the hypothesis, it is clear thatP 6= M. So, ηMP 6= 1. Letr, s ∈ R−pr such thatr · s � ηMP . Evaluating this inM and using [13, Proposition14 (2)], we obtainr(M)Ms(M) ≤ ηMP (M). Thus,r(M)Ms(M) ≤ ηMP (M) ≤ P.so, by hypothesis, it follows thatr(M) ≤ P or s(M) ≤ P. And by Proposition4.9, ηMP (M) is the largest fully invariant submodule ofM which is contained inP.Hence,r(M) ≤ ηMP (M) or s(M) ≤ ηMP (M). Consequently,r � ηMP or s � ηMP .

(2)⇒ (1) By hypothesis,ηMP is a prime preradical, so in particularP � M. LetN,L ∈ Λfi(M) such thatNML = αM

N (L) ≤ P. Notice thatNML = αMN (L) =

(αMN · αM

L )(M). Then,αMN · αM

L � ωMNML. Thus, by Remark4.7 (2) we getαM

N ·

αML � ηMP . SinceηMP is prime, thenαM

N � ηMP or αML � ηMP . Applying M we

obtainN = αMN (M) ≤ ηMP (M) ≤ P or L = αM

L (M) ≤ ηMP (M) ≤ P. ByRemark4.7 (2), we conclude thatr � ηMP or s � ηMP . Hence,ηMP is a primepreradical. �

Corollary 4.11. LetM ∈ R−Mod andP ≤ M. If P is a large prime submoduleof M, thenηMP (M) is the largest element inSpec(Λfi(M)) which is contained inP.

Proof. It follows from Propositions4.9and4.10. �

Definition 4.12. Following [13], anR-moduleM is aprimemodule if

0 ∈ Spec(Λfi(M)).

Corollary 4.13. LetM ∈ R−Mod andP ≤ M. Then, the following conditionshold.

(1) If P is a large prime submodule ofM, thenM/ηMP (M) is a prime module.(2) If M is a quasi-projective module andM/P is a prime module, thenP is

a large prime submodule.

Proof. (1) It is a consequence from Corollary4.11and [13, Proposition 18.1].(2) Note that in [13, Proposition 18.2] is not necessary the hypothesis ofP ∈

Λfi(M). �

Example 4.14. If M < M is a maximal submodule thenM/M is a simple mod-ule, hence it is a prime module. Therefore, every maximal submodule ofM is alarge prime submodule.

Definition 4.15. An R-moduleM is FI-simple if the only fully invariant submod-ules ofM are0 andM .


Proposition 4.16. If M is an FI-simple module, thenM is cogenerated by all itsnon-zero factors modules.

Proof. If M is FI-simple, it is clear thatLgSpec(M) = {N ≤ M | N 6= M}andSpec(Λfi(M)) = 0. By corollary 4.11, ηMN (M) = 0 for all N ≤ M, withN 6= M . This implies that there exists a monomorphismM → (M/N)S , whereS = EndR(M) for all proper submodule ofM . �

Remark 4.17. In [1] the author writesSpecfp(M) instead ofSpec(Λfi(M)).If M is projective inσ[M ] thenLgSpec(M) is a topological space by Proposi-tion 3.18 and Corollary4.13. We can see thatSpec(Λfi(M)) is a subspace ofLgSpec(M). Recall that anR−module isduo if Λ(M) = Λfi(M). Note that ifM is a duo module thenSpec(Λfi(M)) = LgSpec(M) = Spec(Λ(M)). We willcall LgSpec(M) the large spectrumof M .

Definition 4.18. A moduleM is coatomicif every submodule is contained in amaximal submodule ofM .

Example 4.19.(1) Every finitely generated and every semisimple module is coatomic.(2) Semiperfect modules are coatomic.(3) If R is a left perfect ring every leftR-module is coatomic. (see [8])

Note that in general, it could be thatLgSpec(M) = ∅. If we assume thatM isa coatomic module thenLgSpec(M) is not the empty set.

We will denoteSpec(Λ(M)) just bySpec(M).

Proposition 4.20. Let M be quasi-projective. ThenSpec(Λfi(M)) is a densesubspace ofLgSpec(M)

Proof. Let U(N) 6= ∅ be an open set ofLgSpec(M) such that

U(N) ∩ Spec(Λfi(M)) = ∅.

Thus, the elements ofU(N) are not fully invariant. LetP ∈ U(N). By Corollary4.11, there existsQ ∈ Spec(Λfi(M)) such thatQ ⊆ P , which is a contradiction.ThusU(N) = ∅. So,Spec(Λfi(M)) is dense inLgSpec(M). �

Let O(LgSpec(M)) be the frame of open subsets ofLgSpec(M). Then wehave a morphism of

∨

-semilattices

U : Λfi(M) → O(LgSpec(M))

given byU(N) = {P ∈ LgSpec(M) | N * P}.This morphism has a right adjunctU∗ : O(LgSpec(M)) → Λfi(M) given by

U∗(A) =∑

{K ∈ Λfi(M) | U(K) ⊆ A}.

Proposition 4.21.LetN ∈ Λfi(M). Then(U∗◦U)(N) is the largest fully invariant

submodule ofM contained in⋂

P∈V(N)

P.

Proof. It follows by Proposition3.20. �


Definition 4.22. Let M be anR-module andM 6= N ∈ Λfi(M). We say thatNis semiprime inM if wheneverKMK ≤ N with K ∈ Λfi(M) thenK ≤ N . Wesay thatM is asemiprimemodule if0 is semiprime.

Remark 4.23. The last definition is given in [14]. Note that if{Pi}I is a familyof large primes submodules ofM , then

⋂

I

Pi is not necessary a fully invariant

submodule ofM but⋂

I

Pi satisfies the property that for everyL ∈ Λfi(M) such

thatLML ≤⋂

I

Pi thenL ≤⋂

I

Pi.

Proposition 4.24. Let µ = U∗ ◦ U be as in Proposition4.21. LetN ∈ Λfi(M)then,µ(N) = N if and only ifN is semiprime inM or N = M .

Proof. Assume thatµ(N) = N and letL ∈ Λfi(M) such thatLML ≤ N.

We have thatµ(N) ≤⋂

P∈V(N)

P , soLML ≤⋂

P∈V(N)

P . By Remark4.23, L ≤

⋂

P∈V(N)

P . SinceL is a fully invariant submodule ofM , by Proposition4.21

L ≤ µ(N). Thusµ(N) is semiprime inM .Now suppose thatN is semiprime inM . By [6, Proposition 1.12],

N =⋂

{Q | N ≤ Q Q ∈ Spec(Λfi(M))}.

SinceSpec(Λfi(M))⊆LgSpec(M), then⋂

P∈V(N)

P ⊆⋂

{Q ∈ Spec(Λfi(M)) | N ≤ Q} = N.

This implies thatN = µ(N). �

Corollary 4.25. The closure operatorµ : Λfi(M) → Λfi(M) is a multiplicativepre-nucleus.

Proof. By Theorem3.21. �

Remark 4.26. By definitionµ is a closure operator and by Corollary4.25µ is apre-nucleus thenµ is a nucleus inΛfi(M).

Proposition 4.27. LetM ∈ R−Mod and

SP (M) = {N ∈ Λfi(M) | N is semiprime} ∪ {M}.

ThenSP (M) is a frame. Moreover,SP (M) ∼= O(LgSpec(M)) canonically asframes.

Proof. By Proposition4.24, Λfi(M)µ = SP (M). Sinceµ is a multiplicativenucleus thenSP (M) is a frame by Corollary3.11. �

Definition 4.28. Let L be a lattice. An element1 6= p ∈ L is called∧-irreducibleif wheneverx ∧ y ≤ p for anyx, y ∈ L thenx ≤ p or y ≤ p.


Given a frameF, its pointsis the set

pt(F ) = {p ∈ F | p is∧ -irreducible}

Proposition 4.29. LetM be anR-module. Thenpt(SP (M)) = Spec(Λfi(M))

Proof. It is clear thatSpec(Λfi(M)) ⊆ pt(SP (M)). Now, letP ∈ pt(SP (M))andN,L ∈ Λfi(M) such thatNML ≤ P . By Proposition4.25,

µ(N) ∩ µ(L) = µ(N ∩ L) = µ(NML) ≤ µ(P ) = P.

Sinceµ(N), µ(L) ∈ SP (M) thenN ≤ µ(N) ≤ P orL ≤ µ(L) ≤ P . �

Remark 4.30. Sincept(SP (M)) = Spec(Λfi(M)), there exists a continuousfunction

η : LgSpec(M) → Spec(Λfi(M))

defined asη(P ) =∑

SP{N ∈ SP (M) | P ∈ V(N)}, where

∑

SPdenotes the suprema

in the frameSP (M). Also, we have a frame isomorphism betweenSP (M) andO(Spec(Λfi(M))). See [21]

Definition 4.31. LetF be a frame. It is said thatF is spatial if it is isomorphic toO(X) for some topological spaceX. If each quotient frame ofF is spatial, it issaid thatF is totally spatial.

Theorem 4.32. Let M be projective inσ[M ]. Suppose for every fully invariantsubmoduleN ≤ M , the factor moduleM/N has finite uniform dimension. Then,the frameSP (M) is totally spatial.

Proof. Let N ∈ SP (M). By [22, Lemma 9]M/N is projective inσ[M/N ].SinceN is semiprime inM thenM/N is a semiprime module (Definition4.22)by [14, Proposition 13]. By hypothesisM/N has finite uniform dimension, soby [6, Proposition 1.29]Spec(Λfi(M/N)) has finitely many minimal elements(P1/N), ..., (Pn/N) such that0 = P1/N ∩ ... ∩ Pn/N .

SincePi/N ∈ Spec(Λfi(M/N)) thenM/Pi∼=

M/NPi/N

is a prime module. Thus,

by [13, Proposition 18],Pi ∈ Spec(Λfi(M)) for all 1 ≤ i ≤ n. Moreover,N = P1 ∩ ... ∩ Pn. Because of this intersection is finite, we can assume that it isirredundant. Thus by [10, Theorem 3.4],SP (M) is totally spatial. �

For the definition of aweakly scattered space, see [21] and [11, 8.1].

Corollary 4.33. Let M be projective inσ[M ]. Suppose for every fully invariantsubmoduleN ≤ M , the factor moduleM/N has finite uniform dimension. Thenthe topological spaceLgSpec(M) is weakly scattered.

Proof. By Proposition4.27 SP (M) ∼= O(LgSpec(M)), thenLgSpec(M) isweakly scattered by [10, Corollary 3.7]. �

For the definition of Krull dimension of a moduleM see [5].

Corollary 4.34. Let M be projective inσ[M ]. If M has Krull dimension, thenLgSpec(M) is weakly scattered.

Proof. If M has Krull dimension then every factor moduleM/N so does. Now,if M/N has Krull dimension, by [5, Proposition 2.9]M/N has finite uniformdimension. So by Corollary4.33LgSpec(M) is weakly scattered. �


Remark 4.35. Note that with the last corollary, for every commutative ringR withKrull dimension,Spec(R) is weakly scattered. In particular, for every noetheriancommutative ring. Also see [10, Theorem 4.1].

Remark 4.36. In [21, Theorem 7.5], for aT0 spaceS, the frame of all nuclei overthe topology ofS is boolean precisely whenS is scattered, so by4.33, we observethatNO(LgSpec(M)) ∼= NSP (M) is boolean.

If R is a ring, the lowest radical ofR is defined as

Nil∗(R) =⋂

{P | P ∈ Spec(Bil(R))}.

It is well known thatNil∗(R) is contained inRad(R).

Definition 4.37. LetM ∈ R−Mod. The lowest radical ofM is the fully invariantsubmodule

Nil∗(M) =⋂

{Q | Q ∈ Spec(Λfi(M))}.

Proposition 4.38.Nil∗(M) ≤ Rad(M).

Proof. Since every maximal submodule is a large prime submodule by Remark4.23, Rad(M) is semiprime inM . SoRad(M) is an intersection of elements ofSpec(Λfi(M)) by [6, Proposition 1.12]. This implies thatNil∗(M) ≤ Rad(M).

�

IfMax(M) = {M < M | M is a maximal submodule},

we have thatMax(M) ⊆ LgSpec(M). Notice thatMax(M) is a subspace ofLgSpec(M).

Suppose thatM is coatomic. Then we have the adjunction

Λfi(M)m ..

O(Max(M))m∗

mm

defined asm(N) = {M ∈ Max(M) | N * M},

andm∗(A) =

∑

{K ∈ Λfi(M) | m(K) ⊆ A}.

This adjunction can be factorized as

Λfi(M)U

..

m

""

O(LgSpec(M))U∗

mm

I

��

O(Max(M))

m∗

bb

I∗

LL,

whereI : O(LgSpec(M)) → O(Max(M)) is given byI(A) = A ∩Max(M).


Following the proof of Proposition3.20, (m∗◦m)(N) is the largest fully invariantsubmodule contained in

⋂

M∈V(N)∩Max(M)

M

Then(m∗ ◦ m)(0) =

⋂

M∈Max(M)

M = Rad(M)

Note that the proof of Theorem3.21can be applied toτ := m∗ ◦ m, soτ is amultiplicative nucleus. HenceR(M) := Λfi(M)τ is a frame by Corollary3.11.Moreover,R(M) is a subframe ofSP (M).

Definition 4.39. Let M be anR-module. M is co-semisimpleif every simplemodule inσ[M ] isM -injective.

If M = RR, this is the definition of aleft V -ring.

The following characterization of co-semisimple modules is given in [23, 23.1]

Proposition 4.40. For anR-moduleM the following statements are equivalent:(1) M is co-semisimple.(2) Any proper submodule ofM is an intersection of maximal submodules.

Corollary 4.41. LetM be a co-semisimple module. ThenΛfi(M) is a frame.

Proof. SinceM is co-semisimple thenSP (M) = Λfi(M). �

Corollary 4.42. LetM be a duo co-semisimple module thenΛ(M) is a frame.

Proof. In this caseΛ(M) = Λfi(M). �

ACKNOWLEDGMENTS

The authors wish to thank Professor Harold Simmons for sharing [17] whichinspired the topics in this paper. Finally, the authors are grateful for the referee’smany valuable comments and helpful suggestions which have improved this one.

REFERENCES

[1] Abuhlail, J. A Zariski topology for modules.Communications in Algebra, 39 (2011): 4163-4182.

[2] Beachy, J.M -injective modules and primeM -ideals. Communications in Algebra, 30(10),4639-4676 (2002).

[3] Bican, L., Jambor P., Kepka T., Nemec P.Prime and coprime modules.Fundamenta Mathemat-icae, 107:33-44 (1980).

[4] Perez, J. C., & Montes, J. R. (2012).Prime Submodules and Local Gabriel Correspondence inσ[M ]. Communications in Algebra, 40(1), 213-232.

[5] Castro, J., Rıos, J.Krull dimension and classical Krull dimension of modules. Communicationsin Algebra, 42:7, 3183-3204 (2014).

[6] Castro, J., Medina, M., Rıos, J., Zaldıvar, A.On semiprime Goldie Modules, submitted.[7] Golan J.Linear topologies on a ring: an overview, Pitman Research Notes in Mathematics

Series 159, Longman Scientific & Technical, Harlow, 1987, John Wiley & Sons Inc., NewYork.

[8] Gungooroglu, C.Coatomic Modules. Far East J. Math. Sci. Special Volume, Part II (1998):153-162.

[9] Lopez, A.F., Tocon Barroso, M.I.Pseudocomplmented Semilattices, Boolean Algebras, andCompatible ProductsJournal of Algebra 242, 60-91 (2001).


[10] Niefield, S. B., Rosenthal K. I.Spatial sublocales and essential primes.Topology and its Ap-plications 26.3 (1987): 263-269.

[11] Picado, J., A. Pultr.Frames and locales: Topology with our points, Springer Science & BusinessMedia (Frontiers in Mathematics), 2012.

[12] Raggi F., Rıos J., Fernandez-Alonso R., Rincon H., Signoret C.The lattice structure of prerad-icals. Communications in Algebra, 30(3): 533-1544, (2002).

[13] Raggi F., Rıos J., Fernandez-Alonso R., Rincon H., Signoret C.Prime and irreducible preradi-cals.Journal of Algebra and its Applications 4.04 (2005): 451-466.

[14] Raggi F., Rıos J., Fernandez-Alonso R., Rincon H., Signoret C.Semiprime preradicals. Com-munications in Algebra, 37(8): 2811-2822, (2009).

[15] Raggi F., Rıos J., Rincon H., Fernandez-Alonso R.Basic preradicals and main injective mod-ules.Journal of Algebra and Its Applications. Vol. 8, No. 1 (2009)1-16.

[16] Rosenthal K. I.Quantales and Their Applications.Pitman Research Notes in Mathematics Se-ries, Longman Higher Education (1990).

[17] Simmons, H. ”Two sided multiplicative lattices and ring radicals.” preprint.[18] Simmons, Harold.An algebraic version of Cantor-Bendixson analysis.Categorical Aspects of

Topology and Analysis. Springer Berlin Heidelberg, 1982. 310-323.[19] Simmons, H.A collection of notes on frames.http://www.cs.man.ac.uk/˜hsimmons/FRAMES/frames[20] Simmons, H.An introduction to idioms.Notes.http://www.cs.man.ac.uk/˜hsimmons/00-IDSandMODS/EARLIER-VERSIONS/1-IDIOMS.pdf[21] Simmons, Harold. The fundamental triangle of a space.

http://www.cs.man.ac.uk/˜hsimmons/FRAMES/D-FUNDTRIANGLE.pdf

(2006).[22] van der Berg, J., Wisbauer, R.Modules whose hereditary pretorsion classes are closed under

products.Journal of Pure and Applied Algebra 209(1): 215-221, (2007).[23] Wisbauer R. Foundations of Module and Ring Theory. 1991.

http://www.math.uni-duesseldorf.de/˜wisbauer/book.pdf

MAURICIO MEDINA BARCENAS(∗), ANGEL ZALD IVAR CORICHI†.Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, area de Investigacin Cientıfica,Circuito Exterior, C.U., 04510, Mexico, D.F., Mexico.

MARTHA L IZBETH SHAID SANDOVAL M IRANDA (‡).Facultad de Ciencias, Universidad Nacional Autonoma de M´exico.Circuito Exterior, Ciudad Universitaria, 04510, Mexico,D.F., Mexico.e-mails: (∗) [email protected] author

(†) [email protected](‡) [email protected]

http://www.cs.man.ac.uk/~hsimmons/FRAMES/frames

http://www.cs.man.ac.uk/~hsimmons/00-IDSandMODS/EARLIER-VERSIONS/1-IDIOMS.pdf

http://www.cs.man.ac.uk/~hsimmons/FRAMES/D-FUNDTRIANGLE.pdf

http://www.math.uni-duesseldorf.de/~wisbauer/book.pdf

a generalization of quantales with applications to … · martha lizbeth shaid sandoval-miranda...

Documents