International Journal of Computational Intelligence Systems

Volume 13, Issue 1, 2020, Pages 794 - 801

New Kind of MV-Modules

Authors
S. Saidi Goraghani1, R.A. Borzooei2, ORCID, S.S. Ahn3, *, ORCID, Y.B. Jun2, 4
1Department of Mathematics, Farhangian University, Tehran, Iran
2Department of Mathematics, Shahid Beheshti University, Tehran, Iran
3Department of Mathematics Education, Dongguk University, Seoul, Korea
4Department of Mathematics Education, Gyeongsang National University, Jinju, Korea
*Corresponding author. Email: sunshine@dongguk.edu
Corresponding Author
S.S. Ahn
Received 10 March 2020, Accepted 31 May 2020, Available Online 18 June 2020.
DOI
10.2991/ijcis.d.200602.001How to use a DOI?
Keywords
MV-algebra; PMV-algebra; Ak-module; Free Ak-module; Radical of an Ak-module; Noetherian Ak-module
Abstract

In this paper, by considering the notion of MV-modules, which is the structure that naturally correspond to lu-modules over lu-rings, we investigate some properties of a new kind of MV-modules, that we introduced in Borzooei and Saidi Goraghani, Free MV-modules, J. Intell. Fuzzy Syst. 31 (2016), 151–161 as Ak-modules. With the current situation, it was not easy for us to work on some concepts such as free MV-modules and Noetherian MV-modules. So we limited our scope of work by introducing a new kind of MV-modules. We define and study the notions of free Ak-modules, radical of Ak-modules and Noetherian Ak-modules, where A is a product MV-algebra and k. For example, we state a general representation for a free Ak-module, and we obtain conditions in which an Ak-module can be Noetherian.

Copyright
© 2020 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

MV-algebras were defined by Chang [1] as algebras corresponding to the Lukasiewicz infinite valued propositional calculus. These algebras have appeared in the literature under different names and polynomially equivalent presentation: CN-algebras, Wajsberg algebras, bounded commutative BCK-algebras and bricks. It is discovered that MV-algebras are naturally related to the Murray-von Neumann order of projections in operator algebras on Hilbert spaces and that they play an interesting role as invariant of approximately finite-dimensional C-algebras. They are also naturally related to Ulam’s searching games with lies. MV-algebras admit a natural lattice reduction and hence a natural order structure. Many important properties can be derived from the fact, established by Chang that nontrivial MV-algebras are subdirect products of MV-chains, that is, totally ordered MV-algebras. To prove this fundamental result, Chang introduced the notion of prime ideal in an MV-algebra. The categorical equivalence between MV-algebras and lu-groups leads to the problem of defining a product operation on MV-algebras, in order to obtain structures corresponding to l-rings. A product MV-algebra (or PMV-algebra, for short) is an MV-algebra which has an associative binary operation “.”. It satisfies an extra property which will be explained in Preliminaries. During the last years, PMV-algebras were considered and their equivalence with a certain class of l-rings with strong unit was proved. It seems quite natural to introduce modules over such algebras, generalizing the divisible MV-algebras and the MV-algebras obtained from Riesz spaces and to prove natural equivalence theorems. Hence, the notion of MV-modules was introduced as an action of a PMV-algebra over an MV-algebra by Di Nola [2]. Since then, we and some researches have worked on MV-modules (see, for instance, [312]). Most published papers are about ideals of MV-modules. We were surious to investigate concepts such as injective (projective) MV-modules, free MV-modules and so on. In 2018, we introduced injective MV-modules (see [8]), but with the current situation, it was not possible for us to achieve all our goals. So we first limited our scope of work by introducing a new kind of MV-modules, namely Ak-modules. Since MV-modules are in their infancy, stating and opening of any subject in this field can be useful. Hence, in this paper, we present definitions of free Ak-modules, radical of Ak-modules and Noetherian Ak-modules, where A is a PMV-algebra and k. In Section 3, we state when an Ak-module is equal to j=1nA, for some n. In Section 4, we state a method to obtain the radical of an A-ideal in Ak-modules. In Section 5, we obtain conditions in which an Ak-module can be Noetherian. We hope that we have taken an effective step in this regards.

2. PRELIMINARIES

In this section, we review related lemmas and theorems that we use in the next sections.

Definition 2.1.

[13] An MV-algebra is a structure M=(M,,,0) of type (2,1,0) such that

(MV1) (M,,0) is an Abelian monoid,

(MV2) (a)=a,

(MV3) 0a=0,

(MV4) (ab)b=(ba)a,

If we define the constant 1=0 and operations and by ab=(ab), ab=ab, then

(MV5) (ab)=(ab),

(MV6) x1=1,

(MV7) (ab)b=(ba)a,

(MV8) aa=1,

for every a,bM. It is clear that (M,,1) is an Abelian monoid. Now, if we define auxiliary operations and on M by ab=(ab)b and ab=a(ab), for every a,bM, then (M,,,0) is a bounded distributive lattice.

An MV-algebra M is a Boolean algebra if and only if the operation is idempotent, that is, xx=x, for every xM. In an MV-algebra M, the following conditions are equivalent: (i) ab=1, (ii) ab=0, (iii) b=a(ba), (iv) cM such that ac=b, for every a,b,cM. For any two elements a,b of the MV-algebra M, ab if and only if a,b satisfy the above equivalent conditions (i)(iv). An ideal of MV-algebra M is a subset I of M, satisfying the following conditions: (I1): 0I, (I2): xy and yI imply xI, (I3): xyI, for every x,yI. In an MV-algebra M, the distance function d:M×MM is defined by d(x,y)=(xy)(yx) which satisfies (i): d(x,y)=0 if and only if x=y, (ii): d(x,y)=d(y,x), (iii): d(x,z)d(x,y)d(y,z), (iv): d(x,y)=d(x,y), (v): d(xz,yt)d(x,y)d(z,t), for every x,y,z,tM. Let I be an ideal of MV-algebra M. We denote xy (xIy) if and only if d(x,y)I, for every x,yM. So is a congruence relation on M. Denote the equivalence class containing x by xI and MI=xI:xM. Then MI,,,0I is an MV-algebra, where xI=xI and xIyI=xyI, for all x,yM. Let M and K be two MV-algebras. A mapping f:MK is called an MV-homomorphism if (H1): f(0)=0, (H2): f(xy)=f(x)f(y) and (H3): f(x)=(f(x)), for every x,yM. If f is one to one (onto), then f is called an MV-monomorphism (MV-epimorphism) and if f is onto and one to one, then f is called an MV-isomorphism.

Lemma 2.2.

[13] In every MV-algebra M, the natural order “” has the following properties:

  1. xy if and only if yx,

  2. if xy, then xzyz, for every zM.

Lemma 2.3.

[13] Let M and N be two MV-algebras and f:MN be an MV-homomorphism. Then the following properties hold:

  1. Ker(f) is an ideal of M,

  2. if f is an MV-epimorphism, then MKerfN.

Definition 2.4.

[14,15] An l-group is an algebra (G,+,,0,,), where the following properties hold:

  1. (G,+,,0) is a group,

  2. (G,,) is a lattice,

  3. xy implies that x+ay+a, for any x,y,a,bG.

A strong unit u>0 is a positive element with property that for any gG there exits nω such that gnu. The Abelian l-groups with strong unit will be simply called lu-groups [16].

The category whose objects are MV-algebras and whose homomorphisms are MV-homomorphisms is denoted by MV. The category whose objects are pairs (G,u), where G is an Abelian l-group and u is a strong unit of G and whose homomorphisms are l-group homomorphisms is denoted by Ug. The functor that establishes the categorial equivalence between MV and Ug is

Γ:UgMV,
where Γ(G,u)=[0,u]G, for every lu-group (G,u) and Γ(h)=h|[0,u], for every lu-group homomorphism h. The above results allows us to consider an MV-algebra, when necessary, as an interval in the positive cone of an l-group. Thus, many definitions and properties can be transferred from l-groups to MV-algebras. For example, the group addition becomes a partial operation when it is restricted to an interval, so we define a partial addition on an MV-algebra M as follows:

x+y is defined if and only if xy and in this case, x+y=xy, for every x,yM. Moreover, if z+xz+y, then xy (see [2]).

Definition 2.5.

[16] An l-ring is a structure (R,+,.,0,), where (R,+,0,) is an L-group such that, for any x,yR,

x0 and y0 implies x.y0.

A product MV-algebra (or PMV-algebra, for short) is a structure A=(A,,.,,0), where (A,,,0) is an MV-algebra and “.” is a binary associative operation on A such that the following property is satisfied: if x+y is defined, then x.z+y.z and z.x+z.y are defined and (x+y).z=x.z+y.z, z.(x+y)=z.x+z.y, for every x,y,zA, where “+” is the partial addition on A. A unity for the product is an element eA such that e.x=x.e=x, for every xA. If A has a unity for product, then e=1. A PMV-algebra A is called commutative, if x.y=y.x, for every x,yA [4]. A PMV-homomorphism is an MV-homomorphism which also commutes with the product operation. A -ideal of A is an ideal I of A such that if aI and bA entail a.bI and b.aI. The set of -ideals of A is denoted by Id(A).

An lu-ring is a pair (R,u), where (R,+,.,0,) is an l-ring and u is a strong unit of R (i.e., u is a strong unit of the underlying l-group) such that u.uu. The category whose objects are pairs (R,u), where R is an l-ring and u is a strong unit of R and whose homomorphisms are l-ring homomorphisms is denoted by UR. The functor that establishes the categorial equivalence between PMV and UR is

Γ:URPMV,
where Γ(R,u)=[0,u]R, for every lu-ring (R,u) and Γ(h)=h|[0,u], for every lu-ring homomorphism h.

Definition 2.6.

[6] Let I be a proper -ideal of A. I is called a -prime ideal of A, if x.yI implies that xI or yI, for any x,yA.

Proposition 2.7.

[4] Let A be a PMV-algebra. Then i=1nA is a PMV-algebra.

Lemma 2.8.

[16] Let A be a PMV-algebra. Then ab implies that a.cb.c and c.ac.b for every a,b,cA.

Definition 2.9.

[2] Let A=(A,,.,,0) be a PMV-algebra, M=(M,,,0) be an MV-algebra and the operation Φ:A×MM be defined by Φ(a,x)=ax, which satisfies the following axioms:

(AM1) If x+y is defined in M, then ax+ay is defined in M and a(x+y)=ax+ay,

(AM2) If a+b is defined in A, then ax+bx is defined in M and (a+b)x=ax+bx,

(AM3) (a.b)x=a(bx), for every a,bA and x,yM.

Then M is called a (left) MV-module over A or briefly an A-module. We say that M is a unitary MV-module if A has a unity for the product and

(AM4) 1Ax=x, for every xM. Moreover, M is called a Boolean A-module if axaya(xy), for every aA and x,yM (see [4]).

Lemma 2.10.

[2] Let A be a PMV-algebra and M be an A-module. Then

  1. 0x=0,

  2. a0=0,

  3. ax(ax),

  4. ax(ax),

  5. (ax)=ax+(1x),

  6. xy implies axay,

  7. ab implies axbx,

  8. a(xy)axay,

  9. d(ax,ay)ad(x,y),

  10. if xIy, then axIay, where I is an ideal of A,

  11. if M is a unitary MV-module, then (ax)=ax+x, for every a,bA and x,yM.

Definition 2.11.

[2] Let A be a PMV-algebra and M be an A-module. Then an ideal NM is called an A-ideal of M if (I4): axN, for every aA and xN. Let N be a proper A-ideal of M. Then N is called a prime A-ideal of M, if amN implies that mN or a(N:M), for any aA and mM, where (N:M)={aA:aMN} (see [6]).

Lemma 2.12.

[4] Let M be an A-module and N be an A-ideal of M. Then

(N:M)={aA:aMN}
is an ideal of A.

Definition 2.13.

[2] Let A be a PMV-algebra and M1, M2 be two A-modules. A map f:M1M2 is called an A-module homomorphism or ( A-homomorphism, for short) if f is an MV-homomorphism and

(H4): f(ax)=af(x), for every xM1 and aA.

Definition 2.14.

[4] Let M1 and M2 be two A-modules. Then the map f:M1M2 is called an A-homomorphism if and only if it satisfies in (H1), (H3), (H4) and

(H2): if x+y is defined in M1, then h(x+y)=h(xy)=h(x)h(y), for every x,yM1, where “+” is the partial addition on M1. If h is one to one (onto), then h is called an A-monomorphism (epimorphism). If h is onto and one to one, then h is called an A-isomorphism and we write M1M2.

Definition 2.15.

[4] Let M be a unitary A-module and there exists k such that i=1naimi(i=1naimi), for every 1nk, aiA and miM. Then M is called an Ak-module. If i=1naimi(i=1naimi), for every n, then M is called an A-module.

Proposition 2.16.

[4] M is an Ak-module if and only if (i=1naimi)=(i=1nmi)i=1naimi, for every aiA, miM and nk.

Definition 2.17.

[4] Let M be an A-module, TM and

M={i=1nxiti:xiA,tiT, for 1inandx1t1++xntn is defined in Mi=1n},
where “ +” is the partial addition (similarly, if T is infinite set, then M can be defined). Then we say that M is generated by T and we set M=T. If |T|<, then M is called a finitely generated A-module. Specially, if M=m, where mM, then M is called a cyclic A-module.

Definition 2.18.

[7] Let M be an A-module. An A-ideal N of M is called a maximal A-ideal of M, if there exist no A-ideal K of M containing N such that NKM. The set of all maximal A-ideals of M is showed by Max(M). Let I be a proper A-ideal in M. The intersection of all maximal A-ideals of M which contain I is called the radical of I and is denoted by Rad(I). Moreover, the intersection all maximal A-ideals of M is showed by Rad(M).

Definition 2.19.

[17] Let M be an A-module. Then

  1. M is called Noetherian if M satisfies the ascending chain condition (ACC): any chain N1N2 of A-ideals of M is stationary.

  2. We say M has maximum condition, if every non-empty family of A-ideals of M has a maximal element.

Note. From now on, in this paper, we let A be a PMV-algebra, M be an MV-algebra and i=1nxi means x1x2xn.

3. FREE Ak-MODULES

In this section, we present definition of free Ak-modules and we obtain some results on them. For example, we state a general representation for a free Ak-module.

Definition 3.1.

Let M be an Ak-module. If M is a free A-module, then M is called a free Ak-module.

Example 3.2.

  1. By Lemma 2.10(d), every free A-module M is a free A1-module.

  2. If A is the Boolean algebra with two elements, then every MV-algebra M is an AN-module. Now, if M is a free A-module, then M is a free AN-module.

  3. Consider L2={0,1}, L4={0,13,23,1}, ab=min{1,a+b}, a=1a and +,,. are ordinary operations in . Then it is routine to show that (L2,,,.,0) is a PMV-algebra and (L4,,,0) is an MV-algebra. Let operation :L2×L4L4 is defined by a·b=a.b, for every aL2 and bL4. Then it is easy to show that L4 is an L2-module. Now, since

    23=1.23,13=1.13,1=1.13+1.23=1.131.23
    we have L4=13,23. It is easy to see that L4 is a free AN-module, where A=L2.

Lemma 3.3.

Let M be an A-module, I be an ideal of A and N be an A-ideal of M. Then

IMN=i=1krimin:riI,miM,nN
is an A-ideal of M.

Proof.

(I1) and (I3) are clear.

(I2): Let mb, where mM and bIMN. Since bIMN, there exist nN, r1,,rkI and m1,,mkM such that b=i=1krimin and so mi=1krimin=1. By Lemma 2.10(d), rimi+rimi is defined and it is clear that ri+ri is defined, too, for every riA, miM and 1ik. Hence,

mi=1kmin=mi=1k(riri)min=mi=1k(ri+ri)min=mi=1k(rimi+rimi)n=mi=1k(rimirimi)n=mi=1krimii=1krimin=1i=1krimi=1
and so by Proposition 2.16,
m=mb=m(mb)=(m(mi=1krimin))=(m(1mi=1krimi1n))=(m(i=1krimi(mni=1kmi)))=(mi=1krimi1)=(mi=1krimi)=(1mi=1krimi0n)=i=1krimi1n(mi=1kmin)=i=1kriminIMN.

(I4): Let aA and i=1kriminIMN. Since by Lemma 2.10(h),

ai=1krimini=1ka(rimi)an=i=1k(a.ri)mian
and since a.riI, for any 1ik, we have i=1k(a.ri)mianIMN and so by (I2), a(i=1krimin)IMN. Therefore, IMN is an A-ideal of M.

Theorem 3.4.

Let A be commutative and M be a cyclic A-module. Then for every A-ideal N of M, there exists an ideal I of A such that N=IM.

Proof.

Since M is a cyclic A-module, there exists mM such that M=m. By Lemma 2.12, (N:M) is an ideal of A and by Lemma 3.3, (N:M)M is an A-ideal of M. Now, we show that N=(N:M)M. It is clear that (N:M)MN. Now, let nN. Then there exists aA such that n=am. Since

aM=am={a(aim):aiA}={(a.ai)m:aiA}={(ai.a)m):aiA}={ai(am):aiA}={ain:aiA}N,
we have a(N:M) and so n(N:M)M. Hence, N(N:M)M and so N=(N:M)M. Now, by considering I=(N:M), we get that N=IM.

Theorem 3.5.

M is a free Ak-module with basis T={t1,,tn} that j=1ntj=1 and nk if and only if Mj=1nA.

Proof.

() Let M be a free Ak-module with a basis T and j=1ntj=1. By Proposition 2.7, j=1nA is a PMV-algebra. Now, let the operation :A × j=1nAj=1nA be defined by a{aj}j=1n={a.aj}j=1n={aaj}j=1n, for every {aj}j=1nj=1nA and aA. It is easy to show that j=1nA is an A-module. Let Φ:j=1nAM be defined by Φ({aj}j=1n)=j=1najtj, for every ajA and tjT. Then it is clear that ϕ is well defined and ϕ(0)=0. If {aj}j=1n+{bj}j=1n is defined in j=1nA, then aj+bj is defined in A and so ajtj+bjtj is defined in M, for every 1jn and tjT. Hence,

ϕ({aj}j=1n+{bj}j=1n)=ϕ({aj+bj}j=1n)=j=1n(aj+bj)tj=j=1n(ajtj+bjtj)=j=1n(ajtjbjtj)=j=1najtjj=1nbjtj=ϕ({aj}j=1n)ϕ({bj}j=1n).

Also,

ϕ(α{aj}j=1n)=ϕ{αaj}j=1n)=j=1n(α.aj)tj=j=1nα(ajtj)=αj=1najtj=αϕ({aj}j=1n).

Finally, for every {aj}j=1nj=1nA,

ϕ(({aj}j=1n))=ϕ({aj}j=1n)=j=1najtj=j=1najtj(j=1ntj)=(j=1najtj)=(ϕ({aj}j=1n)).

Hence, ϕ is an A-homomorphism. Let ϕ({aj}j=1n)=0, for any {aj}j=1nj=1nA. Since T is a linearly independent set, it is easy to show that aj=0, for every 1jn and so Ker(ϕ)=0. It results that ϕ is an A-monomorphism. It is clear that ϕ is an A-epimorphism and so Mj=1nA.

() Let Mj=1nA. We construct a basis for j=1nA. Let θj={ui}i=1n such that

ui=0ifij1ifi=j
We show that K={θj:1jn} is a basis for j=1nA. Let {aj}j=1nj=1nA. Since 1.a=a.1=a, for every aA,
{aj}j=1n={a1,0,,0}{0,a2,0,,0}{0,,0,an}={a1.1,0,,0}{0,a2.1,0,,0}{0,,0,an.1}=a1.{1,0,,0}a2.{0,1,,0}an.{0,,0,1}=a1.θ1a2.θ2an.θn.(it is easy to see thata1θ1++anθnis defined.)

Hence, j=1nA=K. Now, let j=1naj.θj=0. Then it is easy to show that aj=0, for every 1jn. Now, if ϕ:j=1nAM is an A-isomorphism, then {ϕ(θj):1jn} is a basis for M. Moreover, it is clear that j=1nθj=1.

4. RADICAL OF Ak-MODULES

In [7], the radical of an A-ideal in MV-modules had been defined by maximal ideals. In [10], we presented definition of radical of an A-ideal in MV-modules by prime A-ideals. As fundamental result of this section, in an AN-module, we state a method to obtain the radical of an A-ideal.

Definition 4.1.

[10] Let M be an A-module and N be an A-ideal of M. The intersection of all prime A-ideals of M, including N, is called radical of N and it is shown by radM(N) or rad(N). If N is a prime A-ideal of M, then it is clear that rad(N)=N. If there exists no prime A-ideal of M including N, then we let radM(N)=M.

Example 4.2.

Let A={0,1,2,3} and the operations “ ” and “.” on A be defined as follows:

Consider 0=3,1=2,2=1 and 3=0. Then it is easy to show that (A,,,.,0) is a PMV-algebra and (A,,,0) is an MV-algebra. Now, let the operation :A×AA be defined by ab=a.b, for every a,bA. It is easy to show that A is an MV-module on A, I={0,1}, J={0,2} are prime A-ideals of A and {0} is not a prime A-ideal of A. Also, rad(I)={0,1} and rad({0})={0}.

Theorem 4.3.

Let M be a free A-module and P be a -prime ideal of A. Then (PM:M)=P and PM is a prime A-ideal of M.

Proof.

It is clear that P(PM:M). Let X={x1,,xn} be a basis of M, a(PM:M) and aP. Choose xaX. Since axaPMM, we have axa=i=1naixi, where a1x1++anxn is defined. We have axa(aixi)=1, for every 1in. Then by Proposition 2.10 (j), axaaixixi=1 and so by Proposition 2.16, axaaixi(xaxixi)=0. Hence, axa=0, aixa=0 and so a=ai=0, for every 1in. It means that a=1 and so xa=i=1naixi=0, which is a contradiction. Therefore, (PM:M)P and hence (PM:M)=P. Now, let axPM, where aA and xM. Then there exist siA, for 1in such that x=i=1nsixi, where s1x1++snxn is defined. Let aP. Since (a.si)xii=1n(a.si)xi=axPM, a.siP, for every 1in. Since aP, we have siP, for every 1in and so x=i=1nsixiPM.

Theorem 4.4.

Let M be an A-module and L, N be A-ideals of M. Then

  1. Nrad(N),

  2. if LN, then rad(L)rad(N),

  3. rad(rad(N)=rad(N),

  4. rad(NL)rad(N)rad(L).

Proof.

The proof is routine.

Definition 4.5.

Let I be an ideal of A. The intersection of all -prime ideals of A including I is denoted by rA(I) or r(I). If there exists no -prime ideal of A including I, then we let rA(I)=A.

Example 4.6.

In Example 4.2, I={0,1} is a -prime ideal of A and I={0} is not a -prime ideal of A. It is easy to see that r(I)={0,1} and r({0})={0}.

(ii) Let M2() be the ring of square matrixes of order 2 with real elements and let 0 be the matrix with all elements 0. If we define the order relation on components

A=(aij)i,j=1,20 if and only if aij0 for any i,j,
then M2() is an l-ring. If v=12121212, then (M2(),v) is an lu-ring and so A=Γ(M2(),v) is a PMV-algebra. It is easy to see that Id(A)={{0},A} and {0} is not a -prime ideal of A. Then r({0})=A.

Theorem 4.7.

Let A be unital, M be an A-module, N be an A-ideal of M and I be an ideal of A. Then

  1. rad(IM)=rad(r(I)M),

  2. r(N:M)(rad(N):M).

Proof.

(i) By Lemma 3.3, IM is an A-ideal of M. It is clear that Ir(I). Then IMr(I)M and so by Theorem 4.4 (ii), rad(IM)rad(r(I)M). Now, let P be an arbitrary prime A-ideal of M containing IM. Since I(IM:M)(P:M), we have r(I)(P:M) and so r(I)MP. Hence, rad(r(I)M)P. It results that rad(r(I)M)rad(IM).

(ii) Let P be an arbitrary prime A-ideal of M containing N. Since NP, it is easy to see that (N:M)(P:M). Hence, r(N:M)(P:M) and so

r(N:M)NPSpec(M)(P:M)=(NPSpec(M)P:M)=(rad(N):M).

Theorem 4.8.

Let A be commutative, M be an A-module, N be an A-ideal of M and P be a -prime ideal of A. Then AN(P)={mM:cmPMN,cAP} is an A-ideal of M and PMNAN(P).

Proof.

First, we show that AN(P) is an ideal of M. (I1). The proof is clear.

(I2): Let nm and mAN(P), where m,nM. Then there exists cAP such that cmPMN. By Proposition 2.10(f), cncmPMN and so by Lemma 3.3, cnPMN. Hence, nAN(P).

(I3): Let m,nAN(P). Then there exists c1,c2AP such that c1m,c2nPMN. Let c=c1.c2. Since P is a -prime ideal of A, we have cAP. By Propositions 2.10(h) and Lemma 3.3,

c(mn)cmcn=(c1.c2)m(c1.c2)n=c2(c1m)c1(c2n)PMN.

Hence, c(mn)PMN and so mnAN(P).

(I4): Let mAN(P) and aA. Then there exists cAP such that cmPMN. Since c(am)=(c.a)m=(a.c)m=a(cm)PMN, we have amAN(P). Finally, let c=1, for any mPMN. Then cm=1m=mPMN and so mAN(P). Therefore, PMNAN(P).

Theorem 4.9.

Let A be commutative, M be an A-module, N be an A-ideal of M and P be a -prime ideal of A. Then AN(P)=M or AN(P) is a prime A-ideal of M such that P=(AN(P):M).

Proof.

Let AN(P)M. We show that AN(P) is a prime A-ideal of M and P=(AN(P):M). By Theorem 4.8, AN(P) is an A-ideal of M. Let xmAN(P), for any xA and mM. Then there exists cAP such that c(xm)PMN. We show that mAN(P) or x(AN(P):M). If xP, then by Theorem 4.8, xMPMNAN(P). Hence, x(AN(P):M). If xP, then xAP. Since P is a -prime ideal of A, we have c.xAP. On the other hand, c(xm)PMN and so (c.x)mPMN. Hence, mAN(P). Therefore, AN(P) is a prime A-ideal of M. Now, we prove that P=(AN(P):M). Let pP. Then for any mM, pmPMN. Let c=1. Then c(pm)PMN and so pmAN(P). Hence, PMAN(P) and so P(AN(P):M). Now, let q(AN(P):M) such that qP. Since qMAN(P), we have qmAN(P), for any mM. Hence, there exists cAP such that c(qm)PMN and so (c.q)mPMN. Now, since P is a -prime ideal of A, we have c.qAP and so mAN(P). Hence, M=AN(P), which is a contradiction. Then qP and so (AN(P):M)P. Therefore, P=(AN(P):M).

Theorem 4.10.

Let A be commutative and M be an A-module. Then for every A-ideal N of M,

radM(N)={AN(P):Pis a-prime ideal ofA}.

Proof.

Let T={AN(P):Pis a-prime ideal ofA} and mT. Let L be a prime A-ideal of M including N. Hence, by Theorem 2.7, Q=(L:M) is a -prime ideal of A. Since mAN(P), for every -prime ideal P of A, mAN(Q) and so there exists cAQ such that cmQMN=(L:M)MNLLL. Since L is a prime A-ideal of M and cQ=(L:M), mL. Hence TradM(N). Now, let mradM(N) and P be a -prime ideal of A. Hence, mL, where L is every prime A-ideal of M including N. If AN(P)=M, then the proof is complete. Let AN(P)M. By Theorem 4.9, AN(P) is a prime A-ideal of M and P=(AN(P):M). Now, we show that NAN(P). By Theorem 4.8, we have PMNAN(P) and so NAN(P). Since mradM(N), mAN(P). Hence, mT and so radM(N)T. Therefore, radM(N)=T.

5. THE NOETHERIAN Ak-MODULES

In this section, we present definition of a Noetherian Ak-modules and verify some conditions on them.

Definition 5.1.

Let M be an Ak-module. If M is a Noetherian A-module, then M is called a Noetherian Ak-module.

Example 5.2.

  1. If we consider A as A-module, where ab=a.b, for every a,bA, then every ideal in A is an A-ideal of A. Now, if A is a free Ak-module with finite basis, then A is a Noetherian Ak-module.

  2. Every finite Ak-module is a Noetherian Ak-module.

Theorem 5.3.

Let M be an A-module. Then

  1. M is Noetherian if and only if M has maximum condition.

  2. M is Noetherian if and only if N is finitely generated, for any A-ideal N of M.

Proof.

The proof is routine.

Lemma 5.4.

Let M be an A-module and I,J be A-ideals of M. Then

IJ={ab:(aIandbJ)or(aJandbI)}
is an A-ideal of M.

Proof.

(I1) It is clear that 0IJ.

(I2) Let tab, where abIJ, for any tM. W. L. O. G, let aI and bJ. Then tab=1 and so (ta)bJ. It results that (ta)J and so by (MV4), t(at)=a(ta)IJ. Hence, tI or tJ and so t=t0IJ.

(I3) Let a1b1,a2b2IJ. Then

(a1b1)(a2b2)=(a1a2)(b1b2)IJ.

(I4) Let sA and abIJ. Then by Lemma 2.10 (h), s(ab)sasbIJ and so by (I2), s(ab)IJ.

Lemma 5.5.

Let A be commutative, M be an A-module and N be an A-ideal of M. Then

[N:aA]={mM:(aA)mN} is an A-ideal of M, for every aA, where aA={a.b:bA}.

Proof.

(I1): It is clear that 0[N:aA].

(I2): Let m1m2 and m2[N:aA], for any m1,m2M. Then (a.b)m2N, for some bA. Since m1m2, by Lemma 2.10 (f), we have (a.b)m1(a.b)m2N and so (a.b)m1N, for some bA. Hence, m1[N:aA].

(I3): Let m1,m2[N:aA], for any m1,m2M. Since m1m1m2, (ab)m1(ab)(m1m2) and so (ab)m1(ab)m2(ab)(m1m2). Since ((ab)m1)((ab)m2)((ab)m2),

((ab)m1)((ab)m2)+(ab)m2 is defined. Similarly, m1m2+m2 is defined, too. Thus, we have

((ab)m1)((ab)m2)+(ab)m2=(ab)(m1)(ab)m2(ab)(m1m2)=(ab)(m1m2+m2)=(ab)(m1m2)+(ab)m2.

Since + is cancellative, we have ((ab)m1)((ab)m2)(ab)(m1m2). Now, if we set m1m2 instead of m1, then we have

(ab)(m1m2)((ab)m2)(ab)((m1m2)m2)=(ab)(m1m2)(ab)m2.

It follows that

(ab)(m1m2)=(ab)((m1m2)(ab)m2(ab)(m1m2)((ab)m2)(ab)m1(ab)m2(ab)m1N.

Then (ab)(m1m2)N and so m1m2[N:aA].

(I4): Let tA and m[N:aA]. Then (a.b)mN, for some bA. Since (a.b)mN, we have t((a.b)m)N, for some bA. Then (a.b)(tm)=((a.b).t)m=(t.(a.b))mN and so tm[N:aA].

Lemma 5.6.

Let A be commutative, M be a Boolean A2-module and N be an A-ideal of M. Then

  1. aM={am:mM} is an A-ideal of M, for every aA,

  2. if there exists aA such that NaM and [N:aA] are finitely generated, then N is finitely generated,

  3. if N is not finitely generated and it is maximal in all A-ideals of M that are not finitely generated, then N is a prime A-ideal of M.

Proof.

(i) (I1): It is clear that 0aM.

(I2): Let tam, where amaM and tM. Since amm, we get tm and so tm=1. Now, by Proposition 2.16, since

t=tam=t(tam)=(t(tam))=(t(1tam))=(t(am(tm)))=(tam1)=(tam)=am(tm)=amaM,
we get taM.

(I3): Let m1,m2aM, for any m1,m2M. Since M is a Boolean A-module, by Lemma 2.10 (h), am1am2=a(m1m2)aM.

(I4): Let bA and amaM. Then b(am)=(b.a)m=(a.b)m=a(bm)aM.

(ii) By Lemma 5.5 and (i), aM and [N:aA] are A-ideals of M and by Lemma 5.4, NaM is an A-ideal of M, too. Let ϕ:[N:aA]NaM be defined by ϕ(m)=am. It is routine to show that [N:aA]ker(ϕ)NaM. Since [N:aA] is finitely generated, NaM is finitely generated, too. On the other hand, NaMaMNNaM. Now, since NaM is finitely generated, NNaM is finitely generated, too. Hence, N is finitely generated.

(iii) By (ii), the proof is routine.

Theorem 5.7.

Let A be commutative, M be a Boolean A2-module and N be an A-ideal of M. Then M is Noetherian if and only if any prime A-ideal of M is finitely generated.

Proof.

() Let M be a Noetherian A-module. Then by Theorem 5.3 (ii), every prime A-ideal of M is finitely generated.

() Let every prime A-ideal of M is finitely generated. If

T={N:Nis an A-ideal ofM and Nis not finitely generated},
then by Zorn’ s Lemma, T has a maximal element K and so by Lemma 5.6 (iii), K is a prime A-ideal of M that is not finitely generated, which is a contradiction. Hence, T= and so every A-ideal of M is finitely generated. Therefore, M is a Noetherian A-module.

6. CONCLUSION AND FUTURE DIRECTIONS

The categorical equivalence between MV-algebras and lu-groups, leads us to the definition of product operation on MV-algebras, in order to obtain structures corresponding to l-rings. For this reason, A. Di Nola introduced the notion of MV-modules. Since by current definition, it was not easy for us to achieve all our goals, we limited our scope of work. We introduced a particular category of MV-modules as Ak-modules in [4]. Now, in this paper we presented definitions of free Ak-modules, radical of Ak-modules and Noetherian Ak-modules. The obtained results in the last sections encourage us to continue this long way. In fact, there are many questions in this field that should be verified (see advance results in classical modules). Currently, we are working on a different definition for MV-modules, and we plan to continue our studies by new definition. We believe that the fuzzy (graph) structures of this algebras are also interesting (for more guidance, see [18,19,9]).

CONFLICT OF INTEREST

The authors declare that there is no conflict of interest.

AUTHORS' CONTRIBUTIONS

We have same contribution.

ACKNOWLEDGMENTS

The authors are very indebted to the editor and anonymous referees for their careful reading and valuable suggestions which helped to improve the readability of the paper.

REFERENCES

6.F. Forouzesh, E. Eslami, and A. Borumand Saeid, On preime A-ideals in MV-modules, Univ. Politech. Buch. Sci. Bull., Vol. 76, 2014, pp. 181-198.
9.S. Saidi Goraghani and R.A. Borzooei, Prime ⋅-ideals and fuzzy prime ⋅-ideals in PMV-algebras, Ann. Fuzzy Math. Inf., Vol. 12, 2016, pp. 527-538.
10.S. Saidi Goraghani and R.A. Borzooei, Results on prime ideals in PMV-algebras and MV-modules, Ital. J. Pure Appl. Math., Vol. 37, 2017, pp. 183-196.
11.S. Saidi Goraghani and R.A. Borzooei, Decomposition of A-ideals in MV-modules, Ann. Univer. Craiova Math. Comput. Sci. Ser., Vol. 45, 2018, pp. 66-77.
12.S. Saidi Goraghani and R.A. Borzooei, Most results on A-ideals in MV-modules, J. Algebraic Syst., Vol. 5, 2017, pp. 1-13.
16.A. Di Nola and A. Dvurecenskij, Product MV-algebras, Multiple-Valued Logics, Vol. 6, 2001, pp. 193-215.
Journal
International Journal of Computational Intelligence Systems
Volume-Issue
13 - 1
Pages
794 - 801
Publication Date
2020/06/18
ISSN (Online)
1875-6883
ISSN (Print)
1875-6891
DOI
10.2991/ijcis.d.200602.001How to use a DOI?
Copyright
© 2020 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - S. Saidi Goraghani
AU  - R.A. Borzooei
AU  - S.S. Ahn
AU  - Y.B. Jun
PY  - 2020
DA  - 2020/06/18
TI  - New Kind of MV-Modules
JO  - International Journal of Computational Intelligence Systems
SP  - 794
EP  - 801
VL  - 13
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.200602.001
DO  - 10.2991/ijcis.d.200602.001
ID  - Goraghani2020
ER  -