New Kind of MV-Modules

S. Saidi Goraghani; R.A. Borzooei; S.S. Ahn; Y.B. Jun

doi:10.2991/ijcis.d.200602.001

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

New Kind of MV-Modules

Authors

S. Saidi Goraghani¹, R.A. Borzooei²^,, S.S. Ahn³^{, *}^,, 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.200602.001 How 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 A_k-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 A_k-modules, radical of A_k-modules and Noetherian A_k-modules, where A is a product MV-algebra and k∈ℕ. For example, we state a general representation for a free A_k-module, and we obtain conditions in which an A_k-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, [3–12]). 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 A_k-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 A_k-modules, radical of A_k-modules and Noetherian A_k-modules, where A is a PMV-algebra and k∈ℕ. In Section 3, we state when an A_k-module is equal to ∑j=1nA, for some n∈ℕ. In Section 4, we state a method to obtain the radical of an A-ideal in A_k-modules. In Section 5, we obtain conditions in which an A_k-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) 0′⊕a=0′,

(MV4) (a′⊕b)′⊕b=(b′⊕a)′⊕a,

If we define the constant 1=0′ and operations ⊙ and ⊖ by a⊙b=(a′⊕b′)′, a⊖b=a⊙b′, then

(MV5) (a⊕b)=(a′⊙b′)′,

(MV6) x⊕1=1,

(MV7) (a⊖b)⊕b=(b⊖a)⊕a,

(MV8) a⊕a′=1,

for every a,b∈M. It is clear that (M,⊙,1) is an Abelian monoid. Now, if we define auxiliary operations ∨ and ∧ on M by a∨b=(a⊙b′)⊕b and a∧b=a⊙(a′⊕b), for every a,b∈M, 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, x⊕x=x, for every x∈M. In an MV-algebra M, the following conditions are equivalent: (i) a′⊕b=1, (ii) a⊙b′=0, (iii) b=a⊕(b⊖a), (iv) ∃c∈M such that a⊕c=b, for every a,b,c∈M. For any two elements a,b of the MV-algebra M, a≤b 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): 0∈I, (I2): x≤y and y∈I imply x∈I, (I3): x⊕y∈I, for every x,y∈I. In an MV-algebra M, the distance function d:M×M→M is defined by d(x,y)=(x⊖y)⊕(y⊖x) 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(x⊕z,y⊕t)≤d(x,y)⊕d(z,t), for every x,y,z,t∈M. Let I be an ideal of MV-algebra M. We denote x∼y (x≡Iy) if and only if d(x,y)∈I, for every x,y∈M. So ∼ is a congruence relation on M. Denote the equivalence class containing x by xI and MI=xI:x∈M. Then MI,⊕,′,0I is an MV-algebra, where xI′=x′I and xI⊕yI=x⊕yI, for all x,y∈M. Let M and K be two MV-algebras. A mapping f:M→K is called an MV-homomorphism if (H1): f(0)=0, (H2): f(x⊕y)=f(x)⊕f(y) and (H3): f(x′)=(f(x))′, for every x,y∈M. 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:

x≤y if and only if y′≤x′,
if x≤y, then x⊕z≤y⊕z, for every z∈M.

Lemma 2.3.

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

Ker(f) is an ideal of M,
if f is an MV-epimorphism, then MKerf≅N.

Definition 2.4.

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

(G,+,−,0) is a group,
(G,∨,∧) is a lattice,
x≤y implies that x+a≤y+a, for any x,y,a,b∈G.

A strong unit u>0 is a positive element with property that for any g∈G there exits n∈ω such that g≤nu. 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:Ug→MV,

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 x≤y′ and in this case, x+y=x⊕y, for every x,y∈M. Moreover, if z+x≤z+y, then x≤y (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,y∈R,

x≥0 and y≥0 implies x.y≥0.

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,z∈A, where “+” is the partial addition on A. A unity for the product is an element e∈A such that e.x=x.e=x, for every x∈A. 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,y∈A [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 a∈I and b∈A entail a.b∈I and b.a∈I. 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.u≤u. 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:UR→PMV,

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.y∈I implies that x∈I or y∈I, for any x,y∈A.

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 a≤b implies that a.c≤b.c and c.a≤c.b for every a,b,c∈A.

Definition 2.9.

[2] Let A=(A,⊕,.,′,0) be a PMV-algebra, M=(M,⊕,′,0) be an MV-algebra and the operation Φ:A×M→M 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,b∈A and x,y∈M.

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 x∈M. Moreover, M is called a Boolean A-module if ax⊕ay≤a(x⊕y), for every a∈A and x,y∈M (see [4]).

Lemma 2.10.

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

0x=0,
a0=0,
ax′≤(ax)′,
a′x≤(ax)′,
(ax)′=a′x+(1x)′,
x≤y implies ax≤ay,
a≤b implies ax≤bx,
a(x⊕y)≤ax⊕ay,
d(ax,ay)≤ad(x,y),
if x≡Iy, then ax≡Iay, where I is an ideal of A,
if M is a unitary MV-module, then (ax)′=a′x+x′, for every a,b∈A and x,y∈M.

Definition 2.11.

[2] Let A be a PMV-algebra and M be an A-module. Then an ideal N⊆M is called an A-ideal of M if (I4): ax∈N, for every a∈A and x∈N. Let N be a proper A-ideal of M. Then N is called a prime A-ideal of M, if am∈N implies that m∈N or a∈(N:M), for any a∈A and m∈M, where (N:M)={a∈A:aM⊆N} (see [6]).

Lemma 2.12.

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

(N:M)={a∈A:aM⊆N}

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:M1→M2 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 x∈M1 and a∈A.

Definition 2.14.

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

(H′2): if x+y is defined in M1, then h(x+y)=h(x⊕y)=h(x)⊕h(y), for every x,y∈M1, 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 M1≅′M2.

Definition 2.15.

[4] Let M be a unitary A-module and there exists k∈ℕ such that ∑i=1nai′mi≤(∑i=1naimi)′, for every 1≤n≤k, ai∈A and mi∈M. Then M is called an A_k-module. If ∑i=1nai′mi≤(∑i=1naimi)′, for every n∈ℕ, then M is called an Aℕ-module.

Proposition 2.16.

[4] M is an A_k-module if and only if (∑i=1naimi)′=(∑i=1nmi)′⊕∑i=1nai′mi, for every ai∈A, mi∈M and n≤k.

Definition 2.17.

[4] Let M be an A-module, ∅≠T⊆M and

M={∑i=1nxiti:xi∈A,ti∈T, for 1≤i≤nandx1t1+⋯+xntn is defined in M∑i=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 m∈M, 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 N⊊K⊊M. 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

M is called Noetherian if M satisfies the ascending chain condition (ACC): any chain N1⊆N2⊆⋯ of A-ideals of M is stationary.
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 x1⊕x2⊕⋯⊕xn.

3. FREE A_k-MODULES

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

Definition 3.1.

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

Example 3.2.

By Lemma 2.10(d), every free A-module M is a free A1-module.
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.
Consider L2={0,1}, L4={0,13,23,1}, a⊕b=min{1,a+b}, a′=1−a and +,−,. are ordinary operations in ℝ. Then it is routine to show that (L2,⊕,′,.,0) is a PMV-algebra and (L4,⊕,′,0) is an MV-algebra. Let operation ∙:L2×L4→L4 is defined by a·b=a.b, for every a∈L2 and b∈L4. 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=1.13⊕1.23
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 Aℕ-module, I be an ideal of A and N be an A-ideal of M. Then

IM⊕N=∑i=1krimi⊕n:ri∈I,mi∈M,n∈N

is an A-ideal of M.

Proof.

(I1) and (I3) are clear.

(I2): Let m≤b, where m∈M and b∈IM⊕N. Since b∈IM⊕N, there exist n∈N, r1,⋯,rk∈I and m1,⋯,mk∈M such that b=∑i=1krimi⊕n and so m′⊕∑i=1krimi⊕n=1. By Lemma 2.10(d), ri′mi+rimi is defined and it is clear that ri+ri′ is defined, too, for every ri∈A, mi∈M and 1≤i≤k. Hence,

m′⊕∑i=1kmi⊕n=m′⊕∑i=1k(ri⊕r′i)mi⊕n=m′⊕∑i=1k(ri+r′i)mi⊕n=m′⊕∑i=1k(rimi+r′imi)⊕n=m′⊕∑i=1k(rimi⊕r′imi)⊕n=m′⊕∑i=1krimi⊕∑i=1kr′imi⊕n=1⊕∑i=1kr′imi=1

and so by Proposition 2.16,

m=m∧b=m⊙(m′⊕b)=(m′⊕(m′⊕∑i=1krimi⊕n)′)′=(m′⊕(1m′⊕∑i=1krimi⊕1n)′)′=(m′⊕(∑i=1kr′imi⊕(m′⊕n⊕∑i=1kmi)′))′=(m′⊕∑i=1kr′imi⊕1′)′=(m′⊕∑i=1kr′imi)′=(1m′⊕∑i=1kr′imi⊕0n)′=∑i=1krimi⊕1n⊕(m′⊕∑i=1kmi⊕n)′=∑i=1krimi⊕n∈IM⊕N.

(I4): Let a∈A and ∑i=1krimi⊕n∈IM⊕N. Since by Lemma 2.10(h),

a∑i=1krimi⊕n≤∑i=1ka(rimi)⊕an=∑i=1k(a.ri)mi⊕an

and since a.ri∈I, for any 1≤i≤k, we have ∑i=1k(a.ri)mi⊕an∈IM⊕N and so by (I2), a(∑i=1krimi⊕n)∈IM⊕N. Therefore, IM⊕N is an A-ideal of M.

Theorem 3.4.

Let A be commutative and M be a cyclic Aℕ-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 Aℕ-module, there exists m∈M 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)M⊆N. Now, let n∈N. Then there exists a∈A such that n=am. Since

aM=a≺m≻={a(aim):ai∈A}={(a.ai)m:ai∈A}={(ai.a)m):ai∈A}={ai(am):ai∈A}={ain:ai∈A}⊆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 A_k-module with basis T={t1,⋯,tn} that ∑j=1ntj=1 and n≤k if and only if M≅′∑j=1nA.

Proof.

(⇒) Let M be a free A_k-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=1nA→∑j=1nA be defined by a∙{aj}j=1n={a.aj}j=1n={aaj}j=1n, for every {aj}j=1n∈∑j=1nA and a∈A. It is easy to show that ∑j=1nA is an A-module. Let Φ:∑j=1nA→M be defined by Φ({aj}j=1n)=∑j=1najtj, for every aj∈A and tj∈T. 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 1≤j≤n and tj∈T. Hence,

ϕ({aj}j=1n+{bj}j=1n)=ϕ({aj+bj}j=1n)=∑j=1n(aj+bj)tj=∑j=1n(ajtj+bjtj)=∑j=1n(ajtj⊕bjtj)=∑j=1najtj⊕∑j=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)=α∑j=1najtj=αϕ({aj}j=1n).

Finally, for every {aj}j=1n∈∑j=1nA,

ϕ(({aj}j=1n)′)=ϕ({aj′}j=1n)=∑j=1naj′tj=∑j=1naj′tj⊕(∑j=1ntj)′=(∑j=1najtj)′=(ϕ({aj}j=1n))′.

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

(⇐) Let M≅′∑j=1nA. We construct a basis for ∑j=1nA. Let θj={ui}i=1n such that

ui=0ifi≠j1ifi=j

We show that K={θj:1≤j≤n} is a basis for ∑j=1nA. Let {aj}j=1n∈∑j=1nA. Since 1.a=a.1=a, for every a∈A,

{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.θ1⊕a2.θ2⊕⋯⊕an.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 1≤j≤n. Now, if ϕ:∑j=1nA→M is an A′-isomorphism, then {ϕ(θj):1≤j≤n} is a basis for M. Moreover, it is clear that ∑j=1nθj=1.

4. RADICAL OF A_k-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×A→A be defined by a∙b=a.b, for every a,b∈A. 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 Aℕ-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 a∉P. Choose xa∈X. Since axa∈PM⊆M, we have axa=∑i=1naixi, where a1x1+⋯+anxn is defined. We have axa⊕(aixi)′=1, for every 1≤i≤n. Then by Proposition 2.10 (j), axa⊕ai′xi⊕xi′=1 and so by Proposition 2.16, a′xa⊕aixi⊕(xa⊕xi⊕xi′)′=0. Hence, a′xa=0, aixa=0 and so a′=ai=0, for every 1≤i≤n. 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 ax∈PM, where a∈A and x∈M. Then there exist si∈A, for 1≤i≤n such that x=∑i=1nsixi, where s1x1+⋯+snxn is defined. Let a∉P. Since (a.si)xi≤∑i=1n(a.si)xi=ax∈PM, a.si∈P, for every 1≤i≤n. Since a∉P, we have si∈P, for every 1≤i≤n and so x=∑i=1nsixi∈PM.

Theorem 4.4.

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

N⊆rad(N),
if L⊆N, then rad(L)⊆rad(N),
rad(rad(N)=rad(N),
rad(N∩L)⊆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(ℝ) 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=1,2≥0 if and only if aij≥0 for any i,j,

then M2(ℝ) is an l-ring. If v=12121212, then (M2(ℝ),v) is an lu-ring and so A=Γ(M2(ℝ),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 Aℕ-module, N be an A-ideal of M and I be an ideal of A. Then

rad(IM)=rad(r(I)M),
r(N:M)⊆(rad(N):M).

Proof.

(i) By Lemma 3.3, IM is an A-ideal of M. It is clear that I⊆r(I). Then IM⊆r(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)M⊆P. 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 N⊆P, it is easy to see that (N:M)⊆(P:M). Hence, r(N:M)⊆(P:M) and so

r(N:M)⊆⋂N⊆P∈Spec(M)(P:M)=(⋂N⊆P∈Spec(M)P:M)=(rad(N):M).

Theorem 4.8.

Let A be commutative, M be an Aℕ-module, N be an A-ideal of M and P be a ⋅-prime ideal of A. Then AN(P)={m∈M:cm∈PM⊕N,∃c∈A−P} is an A-ideal of M and PM⊕N⊆AN(P).

Proof.

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

(I2): Let n≤m and m∈AN(P), where m,n∈M. Then there exists c∈A−P such that cm∈PM⊕N. By Proposition 2.10(f), cn≤cm∈PM⊕N and so by Lemma 3.3, cn∈PM⊕N. Hence, n∈AN(P).

(I3): Let m,n∈AN(P). Then there exists c1,c2∈A−P such that c1m,c2n∈PM⊕N. Let c=c1.c2. Since P is a ⋅-prime ideal of A, we have c∈A−P. By Propositions 2.10(h) and Lemma 3.3,

c(m⊕n)≤cm⊕cn=(c1.c2)m⊕(c1.c2)n=c2(c1m)⊕c1(c2n)∈PM⊕N.

Hence, c(m⊕n)∈PM⊕N and so m⊕n∈AN(P).

(I4): Let m∈AN(P) and a∈A. Then there exists c∈A−P such that cm∈PM⊕N. Since c(am)=(c.a)m=(a.c)m=a(cm)∈PM⊕N, we have am∈AN(P). Finally, let c=1, for any m∈PM⊕N. Then cm=1m=m∈PM⊕N and so m∈AN(P). Therefore, PM⊕N⊆AN(P).

Theorem 4.9.

Let A be commutative, M be an Aℕ-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 xm∈AN(P), for any x∈A and m∈M. Then there exists c∈A−P such that c(xm)∈PM⊕N. We show that m∈AN(P) or x∈(AN(P):M). If x∈P, then by Theorem 4.8, xM⊆PM⊕N⊆AN(P). Hence, x∈(AN(P):M). If x∉P, then x∈A−P. Since P is a ⋅-prime ideal of A, we have c.x∈A−P. On the other hand, c(xm)∈PM⊕N and so (c.x)m∈PM⊕N. Hence, m∈AN(P). Therefore, AN(P) is a prime A-ideal of M. Now, we prove that P=(AN(P):M). Let p∈P. Then for any m∈M, pm∈PM⊕N. Let c=1. Then c(pm)∈PM⊕N and so pm∈AN(P). Hence, PM⊆AN(P) and so P⊆(AN(P):M). Now, let q∈(AN(P):M) such that q∉P. Since qM⊆AN(P), we have qm∈AN(P), for any m∈M. Hence, there exists c∈A−P such that c(qm)∈PM⊕N and so (c.q)m∈PM⊕N. Now, since P is a ⋅-prime ideal of A, we have c.q∈A−P and so m∈AN(P). Hence, M=AN(P), which is a contradiction. Then q∈P and so (AN(P):M)⊆P. Therefore, P=(AN(P):M).

Theorem 4.10.

Let A be commutative and M be an Aℕ-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 m∈T. 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 m∈AN(P), for every ⋅-prime ideal P of A, m∈AN(Q) and so there exists c∈A−Q such that cm∈QM⊕N=(L:M)M⊕N⊆L⊕L⊆L. Since L is a prime A-ideal of M and c∉Q=(L:M), m∈L. Hence T⊆radM(N). Now, let m∈radM(N) and P be a ⋅-prime ideal of A. Hence, m∈L, 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 N⊆AN(P). By Theorem 4.8, we have PM⊕N⊆AN(P) and so N⊆AN(P). Since m∈radM(N), m∈AN(P). Hence, m∈T and so radM(N)⊆T. Therefore, radM(N)=T.

5. THE NOETHERIAN A_k-MODULES

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

Definition 5.1.

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

Example 5.2.

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

Theorem 5.3.

Let M be an A-module. Then

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

I⊕J={a⊕b:(a∈Iandb∈J)or(a∈Jandb∈I)}

is an A-ideal of M.

Proof.

(I1) It is clear that 0∈I⊕J.

(I2) Let t≤a⊕b, where a⊕b∈I⊕J, for any t∈M. W. L. O. G, let a∈I and b∈J. Then t′⊕a⊕b=1 and so (t′⊕a)′≤b∈J. It results that (t′⊕a)′∈J and so by (MV4), t⊕(a′⊕t)′=a⊕(t′⊕a)′∈I⊕J. Hence, t∈I or t∈J and so t=t⊕0∈I⊕J.

(I3) Let a1⊕b1,a2⊕b2∈I⊕J. Then

(a1⊕b1)⊕(a2⊕b2)=(a1⊕a2)⊕(b1⊕b2)∈I⊕J.

(I4) Let s∈A and a⊕b∈I⊕J. Then by Lemma 2.10 (h), s(a⊕b)≤sa⊕sb∈I⊕J and so by (I2), s(a⊕b)∈I⊕J.

Lemma 5.5.

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

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

Proof.

(I1): It is clear that 0∈[N:aA].

(I2): Let m1≤m2 and m2∈[N:aA], for any m1,m2∈M. Then (a.b)m2∈N, for some b∈A. Since m1≤m2, by Lemma 2.10 (f), we have (a.b)m1≤(a.b)m2∈N and so (a.b)m1∈N, for some b∈A. Hence, m1∈[N:aA].

(I3): Let m1,m2∈[N:aA], for any m1,m2∈M. Since m1≤m1∨m2, (ab)m1≤(ab)(m1∨m2) and so (ab)m1∨(ab)m2≤(ab)(m1∨m2). Since ((ab)m1)⊙((ab)m2)′≤((ab)m2)′,

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

((ab)m1)⊙((ab)m2)′+(ab)m2=(ab)(m1)∨(ab)m2≤(ab)(m1∨m2)=(ab)(m1⊙m2′+m2)=(ab)(m1⊙m2′)+(ab)m2.

Since + is cancellative, we have ((ab)m1)⊙((ab)m2)′≤(ab)(m1⊙m2′). Now, if we set m1⊕m2 instead of m1, then we have

(ab)(m1⊕m2)⊙((ab)m2)′≤(ab)((m1⊕m2)⊙m2′)=(ab)(m1′∧m2)≤(ab)m2.

It follows that

(ab)(m1⊕m2)=(ab)((m1⊕m2)∨(ab)m2≤(ab)(m1⊕m2)⊙((ab)m2)′⊕(ab)m1≤(ab)m2⊕(ab)m1∈N.

Then (ab)(m1⊕m2)∈N and so m1⊕m2∈[N:aA].

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

aM={am:m∈M} is an A-ideal of M, for every a∈A,
if there exists a∈A such that N⊕aM and [N:aA] are finitely generated, then N is finitely generated,
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 0∈aM.

(I2): Let t≤am, where am∈aM and t∈M. Since am≤m, we get t≤m and so t′⊕m=1. Now, by Proposition 2.16, since

t=t∧am=t⊙(t′⊕am)=(t′⊕(t′⊕am)′)′=(t′⊕(1t′⊕am)′)′=(t′⊕(a′m⊕(t′⊕m)′))′=(t′⊕a′m⊕1′)′=(t′⊕a′m)′=am⊕(t′⊕m)′=am∈aM,

we get t∈aM.

(I3): Let m1,m2∈aM, for any m1,m2∈M. Since M is a Boolean A-module, by Lemma 2.10 (h), am1⊕am2=a(m1⊕m2)∈aM.

(I4): Let b∈A and am∈aM. 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, N⊕aM is an A-ideal of M, too. Let ϕ:[N:aA]→N∩aM be defined by ϕ(m)=am. It is routine to show that [N:aA]ker(ϕ)≃N∩aM. Since [N:aA] is finitely generated, N∩aM is finitely generated, too. On the other hand, N⊕aMaM≃NN∩aM. Now, since N⊕aM is finitely generated, NN∩aM 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 A_k-modules in [4]. Now, in this paper we presented definitions of free A_k-modules, radical of A_k-modules and Noetherian A_k-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

1.C.C. Chang, Algebraic analysis of many-valued logic, Trans. Am. Math. Soc., Vol. 88, 1958, pp. 467-490.

2.A. Di Nola, P. Flondor, and I. Leustean, MV-modules, J. Algebra, Vol. 267, 2003, pp. 21-40.

3.A. Boccuto, A. Di Nola, and G. Vitale, Affine representations of l-groups and MV-algebras, Algebra Univers., Vol. 78, 2017, pp. 563-577.

4.R.A. Borzooei and S. Saidi Goraghani, Free MV-modules, J. Intell. Fuzzy Syst., Vol. 31, 2016, pp. 151-161.

5.A. Estrada and Y.A. Poreda, MVW-rigs and product MV-algebras, J. Appl. Non-Classical Logics, Vol. 29, 2019, pp. 78-96.

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.

7.F. Forouzesh, E. Eslami, and A. Borumand Saeid, Radical of A-ideals in MV-modules, Analele Stiintifice ale Universitatii Al l Cuza din Iasi-Mathematica, Vol. LXII, 2016, pp. 33-57.

8.S. Saidi Goraghani and R.A. Borzooei, On injective MV-modules, Bull. Section Logic., Vol. 47, 2018, pp. 283-298.

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.

13.R. Cignoli, I.M.L. D'Ottaviano, and D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic, Dordrecht, Netherlands, 2000.

14.A. Bigard, K. Keimel, and S. Wolfenstein, Groups et Anneaux Réticulés, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Hidelberg, Germany, New York, NY, USA, Vol. 608, 1977.

15.D. Mundici, Interpretation of AF C*, Int. J. Func. Anal., Vol. 65, 1986, pp. 15-63.

16.A. Di Nola and A. Dvurecenskij, Product MV-algebras, Multiple-Valued Logics, Vol. 6, 2001, pp. 193-215.

17.R.A. Borzooei and S. Saidi Goraghani, Noetherian MV-modules, IEEE, in Fuzzy and Intelligent Systems (CFIS), 2015 4th Iranian Joint Congress (Zahedan, Iran), 2015, pp. 1-5.

18.M. Akram, Fuzzy Lie Algebras, (ISFS) Book Series, Springer, Singapore, 2018.

19.A. Dey, A. Pal, and T. Pal, Interval type 2 fuzzy set in fuzzy shortest path problem, Mathematics, Vol. 4, 2016, pp. 62.

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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.200602.001 How to use a DOI?
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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
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International Journal of Computational Intelligence Systems

New Kind of MV-Modules

1. INTRODUCTION

2. PRELIMINARIES

Definition 2.1.

Lemma 2.2.

Lemma 2.3.

Definition 2.4.

Definition 2.5.

Definition 2.6.

Proposition 2.7.

Lemma 2.8.

Definition 2.9.

Lemma 2.10.

Definition 2.11.

Lemma 2.12.

Definition 2.13.

Definition 2.14.

Definition 2.15.

Proposition 2.16.

Definition 2.17.

Definition 2.18.

Definition 2.19.

3. FREE Ak-MODULES

Definition 3.1.

Example 3.2.

Lemma 3.3.

Proof.

Theorem 3.4.

Proof.

Theorem 3.5.

Proof.

4. RADICAL OF Ak-MODULES

Definition 4.1.

Example 4.2.

Theorem 4.3.

Proof.

Theorem 4.4.

Proof.

Definition 4.5.

Example 4.6.

Theorem 4.7.

Proof.

Theorem 4.8.

Proof.

Theorem 4.9.

Proof.

Theorem 4.10.

Proof.

5. THE NOETHERIAN Ak-MODULES

Definition 5.1.

Example 5.2.

Theorem 5.3.

Proof.

Lemma 5.4.

Proof.

Lemma 5.5.

Proof.

Lemma 5.6.

Proof.

Theorem 5.7.

Proof.

6. CONCLUSION AND FUTURE DIRECTIONS

CONFLICT OF INTEREST

AUTHORS' CONTRIBUTIONS

ACKNOWLEDGMENTS

REFERENCES

Cite this article

3. FREE A_k-MODULES

4. RADICAL OF A_k-MODULES

5. THE NOETHERIAN A_k-MODULES