New definition of MV -semimodules

1 Introduction

In 1935, H. S. Vandiver introduced the concept of semirings. Since then the semirings have been studied by many authors (see, for instance, [1, 25]). Nowadays, the theory of idempotent semirings has many applications in other fields, such as discrete mathematics, computer science and languages, linguistic problems, optimization difficulties, discrete event systems, computational difficulties, and so on. We know the theory arising from the substitution of the fields of real and complex numbers with idempotent semirings or semifields.

C.C. Chang [5, 6] defined MV-algebras as algebras corresponding to the Lukasiewicz infinite valued propositional calculus. MV-algebras have equivalent presentation such as CN-algebras, Wajsberg algebras, bounded commutative BCK-algebras and so on. It is proved that MV-algebras are naturally related to the Murray-von Neumann order of projections in operator algebras on Hilbert spaces and that they have an interesting role to play as invariant of approximately finite-dimensional C^*-algebras. Also Ulams searching games with lies is naturally related to MV-algebras. MV-algebras admit a natural order structure (natural lattice reduction). Chang established this fact that nontrivial MV-algebras are subdirect products of MV-chains. That is, MV-algebras are totally ordered and so some essential properties can be derived from this fact. He introduced the notion of prime ideal in MV-algebras to prove this important result. We know that there is a categorical equivalence between MV-algebras and lu-groups. It leads some researches to define a product operation on MV-algebras to obtain structures corresponding to l-rings. A product MV-algebra (PMV-algebra) is an MV-algebra with an addition operation (an associative binary operation “ . ″). PMV-algebras have been widely studied and researched recently. Then, their equivalence with a certain class of l-rings with strong unit was proved. The introduction of modules over such algebras seemed natural for generalizing the divisible MV-algebras and the MV-algebras obtained from Riesz spaces and to prove natural equivalence theorems. Hence, A. Di Nola [9] introduced the notion of MV-modules as an action of a PMV-algebra over an MV-algebra. By presentation of definition of MV-modules, some researches were encouraged for working on MV-modules and related structures (see, for instance, [4, 27–31]).

MV-semirings are an special class of idempotent semirings strictly connected to MV-algebras, the algebraic semantics of Lukasiewicz propositional logic. In particular, in [12], Di Nola and Russo showed that the two aforementioned categories are isomorphic. This fact allows us to import results and techniques from semiring and ring theory into the study of MV-algebras. It is well known that an effective way to study rings is to study the way in which a ring R acts on its modules. There is a connection between an special category of additively idempotent semirings and MV-algebras. This connection was first observed in [10] and eventually enforced in [2]. On the other hand, every MV-algebra has two semirings reducts isomorphic to each other. Also, the category of MV-semirings, defined in [2], is isomorphic to the one of MV-algebras. Since MV-semirings are a branch of semirings, it seems that MV-modules can be defined over MV-semirings. There are many researchers who are interested in modules structures. We think that stating and opening of any subject in this field can be useful. Hence, in this paper, we present a new definition for MV-modules. We use MV-semirings instead of PMV-algebras. In fact, we define MV-modules over MV-semirings. During the last years, MV-modules have been defined over PMV-algebras. Since MV-semirings are an special class of semirings, we hope that new definition of MV-modules help us to explain MV-modules better than the last. For example, in this paper, we study Q-ideals in MV-semirings, quotient MV-semirings and the prime A-ideals in MV-semimodules, and we investigate some properties on MV-semimodules.

2 Preliminaries

In this section, we review the material that we will use in the following sections.

Definition 2.1. [12, 16] A semiring is algebraic structure (S, ∔, . , 0, 1) of type (2, 2, 0, 0) such that;

(i) (S, ∔, 0) is a commutative monoid,

(ii) (S, . , 1) is a monoid,

(iii) “ . ″ distributes over “^∔″ from either side.

We say that S is commutative if k . l = l . k, for every k, l ∈ S. S is called idempotent if it satisfies the equation k∔k = k, for every k ∈ S.

Let ∅ ≠ W ⊆ S. W is called a left ideal of semiring S if it satisfies the following condition:

(1) if k, l ∈ W, then k∔l ∈ W,

(2) if k ∈ W and r ∈ S, then r . k ∈ W. The proper ideal P of S is called a prime ideal of S if t . h ∈ P implies that t ∈ P or h ∈ P, for any t, h ∈ S.

An MV-semiring, is a commutative and additive idempotent semiring (A, ∔, . , 0, 1) such that there exists a map ◊ : A → A that satisfying the following conditions:

(i) t . h = 0 if and only if h ≤ t^◊ (“ ≤ ″ is naturally defined by means of ∔),

(ii) t + h = (t^◊ . (t^◊ . h) ^◊) ^◊,

for every t, h ∈ A.

Definition 2.2. [7] An MV-algebra is an algebra V = (V, ⊕ , ◊ , 0) of type (2, 1, 0) satisfying the following equations

(MV1) (V, ⊕ , 0) is an Abelian monoid,

(MV2) (t^◊) ^◊ = t,

(MV3) 0^◊ ⊕ t = 0^◊,

(MV4) (t^◊ ⊕ h) ^◊ ⊕ h = (h^◊ ⊕ t) ^◊ ⊕ t, for every t, h ∈ V.

If we define the constant 1 = 0^◊ and operations ⊚ and ⊖ by t⊚h = (t^◊ ⊕ h^◊) ^◊ and t ⊖ h = t⊚h^◊, then

(MV5) (t ⊕ h) = (t^◊⊚h^◊) ^◊,

(MV6) t ⊕ 1 =1,

(MV7) (t ⊖ h) ⊕ h = (h ⊖ t) ⊕ t,

(MV8) t ⊕ t^◊ = 1,

for every t, h ∈ V. It is clear that (V, ⊚, 1) is an Abelian monoid. Now, if we define auxiliary operations ∨ and ∧ on V by t ∨ h = (t⊚h^◊) ⊕ h and t ∧ h = t⊚ (t^◊ ⊕ h), for every t, h ∈ V, then (V, ∨ , ∧ , 0) is a bounded distributive lattice.

In an MV-algebra V, the following conditions are equivalent:

(i) t^◊ ⊕ h = 1,

(ii) t⊚h^◊ = 0,

(iii) h = t ⊕ (h ⊖ t),

(iv) ∃s ∈ V such that t ⊕ s = h, for every t, h, s ∈ V.

For any two elements t, h of the MV-algebra V, t ≤ h if and only if t, h satisfy the above equivalent conditions (i) - (iv).

An ideal of MV-algebra V is a subset I of V, satisfying the following conditions:

(I1): 0 ∈ I,

(I2): t ≤ h and h ∈ I imply t ∈ I,

(I3): t ⊕ h ∈ I, for every t, h ∈ I.

A proper ideal I of V is a prime ideal of V if and only if t ⊖ h ∈ I or h ⊖ t ∈ I (or t ∧ h ∈ I implies that t ∈ I or h ∈ I), for every t, h ∈ V. In an MV-algebra V, the distance function f _d  : V × V → V is defined by f _d  (t, h) = (t ⊖ h) ⊕ (h ⊖ t) which satisfies:

(i): f _d  (t, h) =0 if and only if t = h,

(ii): f _d  (t, h) = f _d  (h, t),

(iii): f _d  (t, z) ≤ f _d  (t, h) ⊕ f _d  (h, z),

(iv): f _d  (t, h) = f _d  (t^◊, h^◊),

(v): f _d  (t ⊕ z, h ⊕ k) ≤ f _d  (t, h) ⊕ f _d  (z, k), for every t, h, z, k ∈ V.

Let I be an ideal of MV-algebra V. We denote t ≡  _I h if and only if f _d  (t, h) ∈ I, for every t, h ∈ V. So ≡ _I is a congruence relation on V. Denote the equivalence class containing t by t I and V I = { t I : t ∈ V } . Then ( V I , ⊕ , ◊ , 0 I ) is an MV-algebra, where ( t I ) ◊ = t ◊ I and t I ⊕ h I = t ⊕ h I , for all t, h ∈ V. Let V and K be two MV-algebras. A mapping f : V → K is called an MV-homomorphism if (H1): f (0) =0, (H2): f (t ⊕ h) = f (t) ⊕ f (h) and (H3): f (t^◊) = (f (t)) ^◊, for every t, h ∈ V. If f is one to one (onto), then f is called an MV-monomorphism (epimorphism) and if f is onto and one to one, then f is called an MV-isomorphism (see [8, 32]). A partial addition on MV-algebra V is defined as follows: t + h is defined if and only if t ≤ h^◊ and in this case, t + h = t ⊕ h, for every t, h ∈ V.

Proposition 2.3. [7] Consider V as an MV-algebra. Then

(i) t ≤ h if and only if h^◊ ≤ t^◊;

(ii) t ≤ h imply that t ⊕ z ≤ h ⊕ z and t⊚z ≤ h⊚z, for every t, h, z ∈ V.

Lemma 2.4. [9] Let V be an MV-algebra. Then for every t, h, z ∈ V,

(a) t + 0 = t;

(b) t ∨ h = t + (t^◊⊚h);

(c) if t + h and (t + h) + z are defined, then h + z and t + (h + z) are defined and

       (t + h) + z = t + (h + z);

(d) if t + z ≤ h + z, then t ≤ h;

(e) if t + z = h + z, then t = h,

where + is the partial addition on V.

Proposition 2.5. [7] Consider V as an MV-algebra. Then

(i) t⊚ (h ∨ z) = (t⊚h) ∨ (t⊚z),

(ii) t ⊕ (h ∧ z) = (t ⊕ h) ∧ (t ⊕ z), for every t, h, z ∈ V.

Proposition 2.6. [12] (i) Let A = (A, ⊕ , ◊ , 0) be an MV-algebra. Then (A, ∨ , ⊚, 0, 1) is a semiring.

(ii) Let A = (A, ∔, . , 0, 1) be an MV-semiring. Then (A, ⊕ , ◊ , 0) is an MV-algebra, where t ⊕ h = (t^◊ . h^◊) ^◊, for all t, h ∈ A.

Definition 2.7. [22] The structure B = (B, ★ , 0) of type (2, 0) such that, for all b₁, b₂, b₃ ∈ B:

(BCK1) ((b₁ ★ b₂) ★ (b₁ ★ b₃)) ★ (b₃ ★ b₂) =0;

(BCK2) (b₁ ★ (b₁ ★ b₂)) ★ b₂ = 0;

(BCK3) b₁ ★ b₁ = 0;

(BCK4) 0 ★ b₁ = 0;

(BCK5) b₁ ★ b₂ = b₂ ★ b₁ = 0 implies that b₁ = b₂,

is called a BCK-algebra. The relation b₁ ≤ b₂ which is defined by b₁ ★ b₂ = 0 is a partial order on B with 0 as least element.

Let (B, ★ , 0) be a BCK-algebra. Then B is called bounded, if there exists 1 ∈ B such that b₁ ≤ 1, for every b₁ ∈ B and in this case, we let Nb₁ = 1 ★ b₁. B is said to be commutative, if b₂ ★ (b₂ ★ b₁) = b₁ ★ (b₁ ★ b₂), for all b₁, b₂ ∈ B. In a BCK-algebra B, we let b₁ ∧ b₂ = b₂ ★ (b₂ ★ b₁) and in a bounded BCK-algebra B, we let b₁ ∨ b₂ = N (Nb₁ ∧ Nb₂), for all b₁, b₂ ∈ B. In bounded commutative BCK-algebra B, for any b₁, b₂ ∈ B, b₁ ∨ b₂ is the least upper bound and b₁ ∧ b₂ is the greatest lower bound of b₁, b₂ and so (B, ∨ , ∧) is a bounded lattice.

3 MV-semimodules

In this section, we present a new definition of MV-semimodules, and we state some examples of MV-semimodules.

In [12], the notion of MV-semimodules was introduced as an action of an MV-semiring over an abelian monoid by A. Di Nola and C. Russo. With that definition, they have obtained some interesting results about MV-semimodules over idempotent semifields. Now, we consider an action of an MV-semiring over an MV-algebra and call it new definition of MV-semimodules. With the new definition, we can more easily study the modular structures of MV-algebras. In fact, with this definition, the difference between MV-modules and ordinary modules becomes clearer.

Definition 3.1. Let A = (A, ∔, . , 0, 1) be an MV-semiring, V = (V, ⊕ , ◊ , 0) be an MV-algebra and φ : A × V ⟶ V is defined by φ (t, v) = tv, for any t ∈ A and v ∈ V. If for any t, h ∈ A and v₁, v₂ ∈ V:

(SMV1) if v₁ + v₂ is defined in V, then tv₁ + tv₂ is defined in V and t (v₁ + v₂) = tv₁ + tv₂;

(SMV2) (t∔h) v₁ = tv₁ ⊕ hv₁;

(SMV3) (t . h) v₁ = t (hv₁).

then V is called a (left) MV-semimodule over A or briefly an A-semimodule. We say that V is a unitary A-semimodule if A has a unity 1 _A for the product, that is

(SMV4) 1 _A v₁ = v₁, for every v₁ ∈ V.

Example 2.3. (i) Let A = {e, s, t, u} and the operations “^∔″ and “ . ″ on A be defined as follows: ∔ e s t u e e s t u s s s u u t t u t u u u u u u . e s t u e e e e e s e s e s t e e t t u e s t u Consider the map ◊ : A ⟶ A such that e^◊ = u,  s^◊ = t,  t^◊ = s and u^◊ = e. Then we can easily to show that (A, ∔, . , e, u) is an MV-semiring and (A, ⊕ , ◊ , e) is an MV-algebra, where ⊕ = ∔. Now, let the operation φ : A × A ⟶ A be defined by φ (t, h) = t . h = th, for every t, h ∈ A. We can easily to show that A is an A-semimodule.

(ii) Let A = L₂ = {0, 1}, t∔h = min {1, t + h}, and the map ◊ : L₂ ⟶ L₂ be defined by t^◊ = 1 - t, for every t, h ∈ L₂, where +, - , . are ordinary operations in ℝ . Then it is routine to show that (L₂, ∔, . , 0, 1) is an MV-semiring. Also, let V = L 4 = { 0 , 1 3 , 2 3 , 1 } and operations “^∔″ and “ ◊ ″ be defined on L₄ similar to L₂. Then it is routine to show that (V, ∔, ◊ , 0) is an MV-algebra. Now, let operation φ : A × V ⟶ V be defined by φ (t, h) = t . h = th, for every t ∈ A and h ∈ V. Then we can easily to show that V is an MV-semimodule over A.

Remark 3.3. [12] Let V = (V, ⊕ , ◊ , 0) be an MV-algebra. If t and h are two idempotent elements of V, then t ⊕ h and t⊚h as well. Moreover, we have t ⊕ h = t ∨ h t ⊚ h = t ∧ h t ∨ t ◊ = 1 t ∧ t ◊ = 0 .

Lemma 3.4. Let A = (A, ∔, . , 0, 1) be an MV-semiring. Then

(i) t ≤ s if and only if ∃h ∈ A such that t∔h = s,

(ii) t ≤ h implies that t . s ≤ h . s,

(iii) t . s^◊ ≤ (t . s) ^◊, where t, h, s ∈ A.

Proof. (i) Let t, s ∈ A and ∃h ∈ A such that t∔h = s. Then we have (t^◊ . (t^◊ . h) ^◊) ^◊ = s and so t^◊ . (t^◊ . h) ^◊ = s^◊ . Hence s ◊ . t = t ◊ . ( t ◊ . h ) ◊ . t = t . t ◊ . ( t ◊ . h ) ◊ = 0 and so t ≤ s. The conversely is proved, similarly.

(ii) Let t ≤ h. Then by (i), there exists k ∈ A such that t∔k = h and so (t∔k) . s = t . s∔k . s = h . s. It results that t . s ≤ h . s.

(iii) Since t ≤ 1, by (i), we have t . s^◊ ≤ s^◊. Also, since (t . s) . s^◊ = 0, we have t . s ≤ s and so s^◊ ≤ (t . s) ^◊. Hence t . s^◊ ≤ (t . s) ^◊. □

Theorem 3.5. Every MV-semiring A is a unitary MV-semimodule on itself, where any elements of A is an idempotent element (that is, t . t = t, for any t ∈ A).

Proof. Let A = (A, ∔, . , 0, 1) be an MV-semiring. Then by Proposition 2 (ii), (A, ⊕ , ◊ , 0) is an MV-algebra, where t ⊕ h = (t^◊ . h^◊) ^◊, for every t, h ∈ A. It is routine to see that t∔h = t ∨ h and t . h = t ∧ h, for every t, h ∈ A. Now, if the operation φ : A × A ⟶ A is defined by φ (t, h) = th = t⊚h, for every t, h ∈ A, then A is an A-semimodule:

(SMV1) If h + s is defined in A, then h ≤ s^◊ and so by Lemma 2 (ii) and (iii), th ≤ ts^◊ ≤ (ts) ^◊. It means that th + ts is defined in A. Now, by Remark 3 and Proposition 2 (i), for every t, h ∈ A, we have t ( h + s ) = t ( h ⊕ s ) = t ⊚ ( h ∨ s ) = t ⊚ h ∨ t ⊚ s = t ⊚ h ⊕ t ⊚ s = th ⊕ ts . (SMV2) For every t, h, s ∈ A, ( t ∔ h ) ⊚ s = ( t ∔ h ) . s = t . s ∔ h . s = t . s ∨ h . s = t ⊚ s ⊕ h ⊚ s = ts ⊕ hs . (SMV3) and (SMV4) are clear.

Example 3.6. In Example 3(i), A is a unitary MV-semimodule on itself.

Theorem 3.7. Let B be a bounded commutative BCK-algebra. Then B is an MV-semimodule over B.

Proof. Since B is bounded and commutative, we have B = (B, ∨ , N, 0) is an MV-algebra (See [7]). Also, we can easily to see that (B, ∨ , ∧ , N, 0, 1) is an MV-semiring. Now, if the operation φ : B × B ⟶ B is defined by φ (b₁, b₂) = b₁ ∧ b₂, for every b₁, b₂ ∈ B, then B is an MV-semimodule on B.□

Proposition 3.8. Let A = (A, ∔, . , 0, 1) be an MV-semiring, V = (V, ⊕ , ◊ , 0) be an MV-algebra, and V be an A-semimodule. Then

(i) t ≤ h implies that tk ≤ hk,

(ii) tk ∨ hk ≤ (t∔h) k,

(iii) k ≤ l implies that tk ≤ tl,

(iv) (tk) ⊚ (tl) ^◊ ≤ t (k⊚l^◊), for every k, l ∈ V and every t, h ∈ A.

Proof. (i) Let t, h ∈ A, k ∈ V and t ≤ h. Then by Lemma 2(i), there exists s ∈ A such that t∔s = h and so tk ⊕ sk = (t∔s) k = hk. It results that tk ≤ hk.

(ii) By Proposition 2 (ii), (A, ⊕ , ◊ , 0) is an MV-algebra, where t ⊕ h = (t^◊ . h^◊) ^◊, for all t, h ∈ A. Then we have t ∨ h = t ⊕ (h^◊ ⊕ t) ^◊. Hence t ≤ t ⊕ (h^◊ ⊕ t) ^◊ and h ≤ t ⊕ (h^◊ ⊕ t) ^◊. So tx ≤ (t ⊕ (h^◊ ⊕ t) ^◊) k and hk ≤ (t ⊕ (h^◊ ⊕ t) ^◊) k. It results that tk ∨ hk ≤ (t ∨ h) k. Now, we have t ∨ h = t ⊕ ( h ◊ ⊕ t ) ◊ = ( t ◊ . ( h . t ◊ ) ◊ ) ◊ = t ∔ h and so tk ∨ hk ≤ (t∔h) k.

(iii) Since k ≤ l, there exists s ∈ V such that k ⊕ s = l and so by (SAM1), tk ⊕ ts = th. It means that tk ≤ tl.

(iv) By (iii), we have tk ∨ tl ≤ t (k ∨ l). Then ( tk ) ⊚ ( tl ) ◊ ⊕ tl = tk ∨ tl ≤ t ( k ∨ l ) = t ( k ⊚ l ◊ ⊕ l ) Now, since k⊚l^◊ ≤ l^◊, we have t (k⊚l^◊) ≤ tl^◊ ≤ (tl) ^◊. Then t (k⊚l^◊) + tl is defined and so t ( k ⊚ l ◊ ⊕ l ) = t ( k ⊚ l ◊ + l ) = t ( k ⊚ l ◊ ) + tl It results that ( tk ) ⊚ ( tl ) ◊ + tl ≤ t ( k ⊚ l ◊ ) + tl and so by Lemma 2 (d), (tk) ⊚ (tl) ^◊ ≤ t (k⊚l^◊). □

Definition 3.9. Let A = (A, ∔, . , 0, 1) be an MV-semiring, V = (V, ⊕ , ◊ , 0) be an MV-algebra, and V be an A-semimodule. Then an ideal W of V is called an A-ideal of V if (I4): tv ∈ W, for every t ∈ A and v ∈ W. Moreover, W is called a prime A-ideal of V, if tv ∈ W implies that v ∈ W or t ∈ (W : V) = {t ∈ A : tV ⊆ W}, for any t ∈ A and v ∈ V.

Example 3.10. By Example 3 (i), W = {e, s} and T = {e, t} are prime A-ideals of A and E = {e} is not a prime A-ideal of A. Also, I = {e, s} is a prime ideal of A and J = {e} is not a prime ideal of A.

Proposition 3.11. Let A = (A, ∔, . , 0, 1) be an MV-semiring, and W be an ideal of A as an MV-algebra. Then W is an ideal of A.

Proof. By Proposition 2 (ii), we know that (A, ⊕ , ◊ , 0) is an MV-algebra, where t ⊕ h = (t^◊ . h^◊) ^◊, for every t, h ∈ A. We have t ∔ h = ( t ◊ . ( t ◊ . h ) ◊ ) ◊ = t ⊕ ( t ⊕ h ◊ ) ◊ . Since (t ⊕ h^◊) ^◊ ≤ h, we have (t ⊕ h^◊) ^◊ ⊕ t ≤ h ⊕ t and so t∔h ≤ t ⊕ h ∈ W. It results that t∔h ∈ W. Now, let t ∈ W and r ∈ A. Since t . r ≤ t ∈ W, we have t . r ∈ W. Therefore, W is an ideal of A.□

Theorem 3.12. Let A = (A, ∔, . , 0, 1) be an MV-semiring such that k . k = k, for every k ∈ A, and P be an ideal of A. Then

(i) P is an ideal of A as an MV-algebra;

(ii) If k . l^◊, l ∈ P, then k ∈ P, for any k, l ∈ A;

(iii) If P is a prime ideal of A as an MV-semiring, then P is a prime ideal of A as an MV-algebra;

(iv) P is a prime ideal of A if and only if k . l^◊ ∈ P or l . k^◊ ∈ P, for every k, l ∈ A.

Proof (i) By Proposition 2 (ii), we know that (A, ⊕ , ◊ , 0) is an MV-algebra, where t ⊕ h = (t^◊ . h^◊) ^◊, for every t, h ∈ A. Since k . k = k, for every k ∈ A, We can easily to see that t ⊕ h ∈ P, for every t, h ∈ P. Let t ≤ h and h ∈ P, for some t, h ∈ A. We must show that t ∈ P. Since t ≤ h, we have h^◊ . t = 0 ∈ P. On the other hand, by Theorem 3, (SMV1) and (SMV4), we have t = t 1 = t ( h ⊕ h ◊ ) = th ⊕ th ◊ = t . h ⊕ t . h ◊ ∈ P . Hence P is an ideal of A as an MV-algebra.

(ii) Since k . l^◊, l ∈ P, by (i), we have k . l^◊ ⊕ l ∈ P. Also, since k . ( k . l ◊ ⊕ l ) ◊ = k . ( ( k . l ◊ ) ◊ . l ◊ ) = ( k . l ◊ ) . ( k . l ◊ ) ◊ = 0 , we have k ≤ k . l^◊ ⊕ l ∈ P and so k ∈ P.

(iii) Consider P as a prime ideal of A (A is considered as an MV-semiring), and k ∧ l ∈ P, for some k . l ∈ A. We must show that k ∈ P or l ∈ P. Since k . ( k . l ◊ ) ◊ = k . ( k ◊ ⊕ l ) = ( k ◊ ⊕ ( k ◊ ⊕ l ) ◊ ) ◊ = k ∧ l ∈ P , we have k . (k . l^◊) ^◊ ∈ P and so k ∈ P or (k . l^◊) ^◊ ∈ P. Let k ∉ P. Then (k . l^◊) ^◊ ∈ P. Now, we have l ≤ k^◊ ⊕ l = (k . l^◊) ^◊ ∈ P and so by (i), we have l ∈ P.

(iv) (⇒) Consider P as a prime ideal of A. Since (k . l^◊) . (l . k^◊) =0 ∈ P, we have k . l^◊ ∈ P or l . k^◊ ∈ P, for every k, l ∈ A.

(⇐) Let k . l^◊ ∈ P or l . k^◊ ∈ P, for every k, l ∈ A. Now, let t . h ∈ P, t ∉ P and h ∉ P. We consider t . h^◊ ∈ P. Then t . (h∔h^◊) = t . h∔t . h^◊ ∈ P and so h∔h^◊ ∈ P. Now, we have t . ( h ∔ h ◊ ) ◊ = t . ( h ◊ . ( h ◊ . h ◊ ) ◊ ) = ( t . h ◊ ) . ( h ◊ . h ◊ ) ◊ ∈ P and so by (ii), we have t ∈ P that is a contradiction. Hence t ∈ P or h ∈ P.

Note. From now on, in this paper, we let A be an MV-semiring and V be an MV-algebra.

Definition 3.13. [17, 18] Let I be an ideal of A. Then I is called a Q-ideal of A if there exists a subset Q of A such that: (1) A = ⋃ {g∔I : g ∈ Q};

(2) if g₁, g₂ ∈ Q, then (g₁∔I)∩ (g₂∔I) ≠ ∅ if and only if g₁ = g₂.

Now, by considering the notion of Q-ideal on an MV-semiring, we construct the quotient MV-semiring.

Let A I = { g ∔ I : g ∈ Q } and binary operations ⊞ and ⊡ be defined on A I as follows:

(g₁∔I) ⊞ (g₂∔I) = g₃∔I where g₃ ∈ Q is a unique element such that g₁∔g₂∔I⊆ g₃∔I ;

(g₁∔I) ⊡ (g₂∔I) = g₄∔I where g₄ ∈ Q is a unique element such that g₁g₂∔I ⊂ g₄∔I .

Then ( A I , ⊞ , ⊡ , 0 ∔ I ) is an MV-semiring related to I.

Proposition 3.14. Let P be a prime Q-ideal of A. Then A P is a chain MV-semiring.

Proof. Let k, l ∈ A. Then by Theorem 3(iv), we have k . l^◊ ∈ P or l . k^◊ ∈ P and so k . l^◊∔P = 0∔P or l . k^◊∔P = 0∔P. Hence (k∔P) . (l^◊∔P) =0∔P or (l∔P) . (k^◊∔P) =0∔P and so k∔P ≤ l∔P or l∔P ≤ k∔P. Therefore, A P is a chain MV-semiring. □

Note. In the following, we will introduce some general examples for Q-ideals of A. See Lemmas 3 and 3.

Definition 3.15. Let V be an A-semimodule, and W be an A-ideal of V. Then W is called a torsion free A-ideal, if tw = 0 implies that t = 0 or w = 0, where t ∈ A and w ∈ W. Moreover, if V = W, then V is called a torsion free MV-semimodule on A (or torsion free A-semimodule).

Example 3.16. In Example 3 (i), I = {e, s} and J = {e, t} are torsion free A-ideals of A.

(ii) In Example 3 (ii), L₄ is a torsion free MV-semimodule over L₂.

(iii) In Example 3 (i), A is not a torsion free MV-semimodule.

Lemma 3.17. Let V be a unitary A-semimodule, and N be an A-ideal of V. Then

(i) Ann _A  (N) = {r ∈ A : rN = 0} is an ideal of A.

(ii) If N is a torsion free A-ideal of V, then Ann _A  (N) is a Q-ideal of A.

Proof. (i) Let a, b ∈ Ann _A  (N). Then aN = bN = 0. By (SAM2), we have (a∔b) x = ax ⊕ bx = 0, for every x ∈ N. Thus (a∔b) N = 0 and so a∔b ∈ Ann _A  (N). We can easily to see that ar ∈ Ann _A  (N), for every a ∈ A and r ∈ Ann _A  (N). Therefore, Ann _A  (N) is an ideal of A.

(ii) By (i), Ann _A  (N) is an ideal of A. We consider Q = (A - Ann _A  (N)) ∪ {0}, and we show that Ann _A  (N) is a Q-ideal of A. It is clear that ⋃ { q ∔ Ann A ( N ) : q ∈ Q } ⊆ A . Let a ∈ A. If a ∈ Ann _A  (N), then a ∈ 0 ∔ Ann A ( N ) ⊆ ⋃ { q ∔ Ann A ( N ) : q ∈ Q } . If a ∉ Ann _A  (N), then a ∈ Q and so a = a ∔ 0 ∈ ⋃ { q ∔ Ann A ( N ) : q ∈ Q } . Hence A = ⋃ {q∔Ann _A  (N) : q ∈ Q}. Now, we show that ( q 1 ∔ Ann A ( N ) ) ∩ ( q 2 ∔ Ann A ( N ) ) = ∅ , for every q₁, q₂ ∈ Q. Let ( q 1 ∔ Ann A ( N ) ) ∩ ( q 2 ∔ Ann A ( N ) ) ≠ ∅ , for some q₁, q₂ ∈ Q, where q₁ ≠ q₂. We consider two cases:

(1) Let q₁ = 0, q₂ ≠ 0 and there exists r ∈ (q₁∔Ann _A  (N)) ∩ (q₂∔Ann _A  (N)). Then rN = 0, and there exists t ∈ Ann _A  (N) such that r = q₂∔t. It results that q₂N = q₂N∔tN = rN = 0. Since q₂ ∉ Ann _A  (N), we have q₂ = 0 that is a contradiction. Hence q₁ = q₂.

(2) Let q₁, q₂ ≠ 0 and there exists r ∈ ( q 1 ∔ Ann A ( N ) ) ∩ ( q 2 ∔ Ann A ( N ) ) . Then r = q₁∔t₁ = q₂∔t₂, where t₁, t₂ ∈ Ann _A  (N). Now, we have t 1 m = t 2 m = 0 rm = q 1 m ∔ t 1 m = q 2 m ∔ t 2 m , for every m ∈ N. Hence q₁m = q₂m and so q 1 ◊ ( q 2 m ) = q 1 ◊ ( q 1 m ) = ( q 1 ◊ . q 1 ) m = 0 , for every m ∈ N. Since N is torsion free, we have q 1 ◊ = 0 or q₂m = 0, for every m ∈ V. If q₂m = 0, for every m ∈ M, then q₂ ∈ Ann _A  (N) that is a contradiction. So q 1 ◊ = 0 . Similarly, we have q 2 ◊ = 0 and so q 1 ◊ = q 2 ◊ . Therefore, q₁ = q₂ and therefore, Ann _A  (N) is a Q-ideal of A. □

Theorem 3.18. Let V be a unitary torsion free A-semimodule, and P be an A-ideal of V. Then P is a prime A-ideal of V if and only if P is a prime A Ann A ( V ) -ideal of V.

Proof. (⇒) Let P be a prime A-ideal of V. Firstly, we show that V is an A Ann A ( V ) -semimodule. By Lemma 3, Ann _A  (V) is a Q-ideal of A, where Q = (A - Ann _A  (V)) ∪ {0}. If we define operation A Ann A ( V ) × V ⟶ V by (a + Ann _A  (V)) x = ax, for every a ∈ A and x ∈ V, then we have:

(SMV1) : Let x + y be defined in V, for some x, y ∈ V. Since V is an MV-semimodule, ax + ay is defined in V, for every a ∈ A. So (a∔Ann _A  (V)) x + (a∔Ann _A  (V)) y is defined in V and so (a∔Ann _A  (V)) (x + y) = (a∔Ann _A  (V)) x + (a∔Ann _A  (V)) y .

(SMV2) : Let a, b ∈ Q. We have (a∔Ann _A  (V)) ⊞ (b∔Ann _A  (V)) = c∔Ann _A  (V) , where c ∈ Q and a∔b∔Ann _A  (V) ⊆ c∔Ann _A  (V). Since (a∔b∔Ann _A  (V)) ∩ (c∔Ann _A  (V)) ≠ ∅ , we have a∔b = c and so

((a∔Ann _A  (V)) ⊞ (b∔Ann _A  (V))) x = (c∔Ann _A  (V)) x = (a∔b) x = ax ⊕ bx = (a∔Ann _A  (V)) x ⊕ (b∔Ann _A  (V)) x .

(SMV3) : The proof is easy.

Now, let (a∔Ann _A  (V)) x ∈ P, for any a ∈ Q and x ∈ V. Then ax ∈ P and so x ∈ P or a ∈ (P : V). If a ∈ (P : V), then aV ⊆ P ⇔ ax ∈ P , for every x ∈ V ⇔ ( a + Ann A ( V ) ) x ∈ P , for every x ∈ V ⇔ ( a ∔ Ann A ( V ) ) V ⊆ P ⇔ ( a ∔ Ann A ( V ) ) ∈ ( P : V ) s Therefore, x ∈ P or (a∔Ann _A  (V)) ∈ (P : V) and therefore, P is a prime A Ann A ( V ) -ideal of V.

(⇐) The proof is routine. □

Lemma 3.19. Let V be a unitary A-semimodule and N be an A-ideal of V. Then

(i) (N : V) is an ideal of A.

(ii) If N is a prime A-ideal of V such that a^◊m ∈ N implies am ∈ N, for every a ∈ A and m ∈ V, then (N : V) is a Q-ideal of A.

Proof. i) We can easily to see that (N : V) is an ideal of A.

(ii) By (i), (N : V) is an ideal of A. We consider Q = (A - (N : V)) ∪ {0}, and we show that (N : V) is a Q-ideal of A. We can easily to see that A = ⋃ {q∔ (N : V) : q ∈ Q}. Now, we show that (q₁∔ (N : V))∩ (q₂∔ (N : V)) = ∅, for every q₁, q₂ ∈ Q, where q₁ ≠ q₂. Let ( q 1 ∔ ( N : V ) ) ∩ ( q 2 ∔ ( N : V ) ) ≠ ∅ , for some q₁, q₂ ∈ Q, where q₁ ≠ q₂ and there exists r ∈ ( q 1 ∔ ( N : V ) ) ∩ ( q 2 ∔ ( N : V ) ) . We consider two cases:

(1) Let q₁ = 0, q₂ ≠ 0. Then there exists t ∈ (N : V) such that r = q₂∔t, rV ⊆ N, tV ⊆ N and q₂V ⊈ N. So rm, tm ∈ N, for every m ∈ V and there is m₁ ∈ V such that q₂m₁ ∉ N. Now, since q₂m₁ ≤ q₂m₁ ⊕ tm₁ = rm₁ ∈ N, we have q₂m₁ ∈ N that is a contradiction. Hence q₂ = 0 and so q₁ = q₂.

(2) Let q₁, q₂ ≠ 0, where r = q₁∔t₁ = q₂∔t₂, for t₁, t₂ ∈ (N : V). Then q₁V ⊈ N, q₂V ⊈ N and there exists m ∈ V such that q₁m ∉ N. We consider two cases:

(a) If q₂m ∈ N, then q 1 m ≤ q 1 m ⊕ t 1 m = ( q 1 ∔ t 1 ) m = rm = ( q 2 ∔ t 2 ) m = q 2 m ⊕ t 2 m ∈ N and so q₁m ∈ N that is a contradiction. So q₁ = q₂.

(b) Let q₂m ∉ N, too. Then since q₁∔t₁ ≤ q₂∔t₂, we have (q₂∔t₂) ^◊ . (q₁∔t₁) =0 and so q 2 ◊ . ( q 2 ∔ t 2 ) ◊ . ( q 1 ∔ t 1 ) = 0 . It results that q 2 ◊ . ( q 1 ∔ t 1 ) ≤ q 2 ◊ . t 2 and so by Proposition 3 (ii), ( q 2 ◊ . ( q 1 ∔ t 1 ) ) m ≤ ( q 2 ◊ . t 2 ) m = q 2 ◊ ( t 2 m ) ∈ N . Hence ( q 2 ◊ . ( q 1 ∔ t 1 ) ) m ∈ N . Since q 2 ◊ . q 1 ) m ≤ ( q 2 ◊ . q 1 ) m ⊕ ( q 2 ◊ . t 1 ) m = ( q 2 ◊ . q 1 ∔ q 2 ◊ . t 1 ) m = ( q 2 ◊ . ( q 1 ∔ t 1 ) ) m ∈ N , we have q 1 ( q 2 ◊ m ) = ( q 2 ◊ . q 1 ) m ∈ N . Now, since q₁ ∉ (N : V) and N is a prime A-ideal of V, we have q 2 ◊ m ∈ N . It follows that q₂m ∈ N that is a contradiction. Therefore, q₁ = q₂ and therefore, (N : V) is a Q-ideal of A. □

Theorem 3.20. Let V be a unitary A-semimodule and N be a proper A-ideal of V such that a^◊m ∈ N implies am ∈ N, for every a ∈ A and m ∈ V. Then N is a prime A-ideal of V if and only if P = (N : V) is a prime ideal of A and V N is a torsion free A P -semimodule.

Proof. (⇐) Let P = (N : V) be a prime ideal of A and V N be a torsion free A P -semimodule. Then it is routine to show that N is a prime A-ideal of V.

(⇒) Let N be a prime A-ideal of V. The first, we show that V N is an A P -semimodule. By Lemma 3, (N : V) is a Q-ideal of A, where Q = (A - (N : V)) ∪ {0}. Let operation A P × V N ⟶ V N be defined by ( a + P ) x N = ax N , for every a ∈ A and x ∈ V. Then

(SMV1) : If m 1 N + m 2 N is defined in V N , then m 1 N ≤ ( m 2 N ) ◊ and so m₁⊚m₂ ∈ N. By Proposition 3 (v), we have am 1 ⊚ ( am 2 ◊ ) ◊ ≤ a ( m 1 ⊚ m 2 ) ∈ N and so am 1 ⊚ ( am 2 ◊ ) ◊ ∈ N . Hence am 1 N ≤ am 2 ◊ N ≤ ( am 2 ) ◊ N and so am 1 N + am 2 N is defined in V N . Hence ( a ∔ P ) ( m 1 N + m 2 N ) = am 1 N + am 2 N . (SMV2) : Let a, b ∈ Q. We have a∔P⊞b∔P = c∔P, where c ∈ Q and a∔b∔P ⊆ c∔P. Since a∔b∔P∩ c∔P ≠ ∅, we have a∔b = c and so by (SAM2), ( a ∔ P ⊞ b ∔ P ) m N = ( c ∔ P ) m N = ( a ∔ b ∔ P ) m N = ( a ∔ b ) m N = am ⊕ bm N = am N ⊕ bm N = ( a ∔ P ) m N ⊕ ( b ∔ P ) m N (SMV3) : We must show that ( ( a 1 ∔ P ) . ( a 2 ∔ P ) ) m N = ( a 1 ∔ P ) ( ( a 2 ∔ P ) m N ) , for every a₁, a₂ ∈ A and m ∈ V. We have

(a₁∔P) . (a₂∔P) = a₃∔P, where a₃ ∈ Q and a₁ . a₂ + P ⊆ a₃ + P. If a₁ . a₂ ∈ Q, then a₁ . a₂ = a₃ and so ( a 1 ∔ P ) . ( a 2 ∔ P ) ) m N = ( a 1 . a 2 ∔ P ) m N = ( a 1 . a 2 ) m N = a 1 ( a 2 m ) N = ( a 1 ∔ P ) ( ( a 2 ∔ P ) m N ) . Consider a₁ . a₂ ∉ Q. Then a₁ . a₂ ∈ (N : V) and so (a₁ . a₂) V ⊆ N. We have a₁ . a₂∔P ⊆ a₃∔P. Then there are t, k ∈ P such that a₁ . a₂∔t = a₃∔k and so (a₁ . a₂) m + tm = a₃m + km. Since a 3 m ≤ a 3 m + km = ( a 1 . a 2 ) m + tm ∈ N , we have a₃m ∈ N and so a₃ ∈ (V : N) that is a contradiction. Hence a₁ . a₂ ∈ Q.

Finally, we show that V N is a torsion free A P -semimodule. Let ( a ∔ P ) m N = 0 N and a∔P ≠ 0∔P. Then am = f _d  (am, 0) ∈ N. If m N ≠ 0 N , then m = f _d  (m, 0) ∉ N. Since N is a prime ideal of A, we have a ∈ (N : V) = P. Hence a∔P = 0∔P that is a contradiction. Therefore, m N = 0 N . □

4 Prime A-ideals in MV-semimodules

In this section, by the obtained results of section 3, we present some results on prime A-ideals in MV-semimodules.

Lemma 4.1. (i) Let z ∈ A. Then the set I = { x ∈ A : x ≤ z } is an ideal of A (we set I =≺ z ≻).

(ii) Let V be a unitary A-semimodule and m ∈ V. Then I m = { ∑ i = 1 k t i m = t 1 m ⊕ ⋯ ⊕ t k m : ∑ i = 1 k t i m ≤ n ( tm ) , for some n , k ∈ ℕ ∪ { 0 } , where t , t i ∈ A } is an A-ideal of V.

Proof. (i) Let x, y ∈ I. Then x ≤ z and y ≤ z. It results that x + y ≤ z + z = z and so x + y ∈ I. Now, let x ∈ I and a ∈ A. Then x ≤ z and so by Theorem 3 and Proposition 3 (ii), x . a ≤ z . a. Since z^◊ . (z . a) = (z^◊ . z) . a = 0, we have z . a ≤ z and so x . a ≤ z. It results that x . a ∈ I, for every x ∈ I and a ∈ A. Therefore, I is an ideal of A.

(ii) (I₁) It is clear that 0 ∈ I _m .

(I₂) Let x ≤ ∑ i = 1 k t i m ∈ I m , for some x ∈ V. Then x = 1 x ≤ ∑ i = 1 k t i m ≤ n ( tm ) ∈ I m , for some n ≥ 0 and t ∈ A, and so x ∈ I _m .

(I₃) Let ∑ i = 1 k t i m , ∑ i = 1 w s i m ∈ I m . Then there exist n₁, n₂ ≥ 0 and s, t ∈ A such that ∑ i = 1 k t i m ≤ n 1 ( tm ) and ∑ i = 1 w s i m ≤ n 2 ( sm ) and so ∑ i = 1 k t i m ⊕ ∑ i = 1 w s i m ≤ n 1 ( tm ) ⊕ n 2 ( sm ) = tm ⊕ ⋯ ⊕ tm ︸ n 1 times ⊕ sm ⊕ ⋯ ⊕ sm ︸ n 2 times = ( t + ⋯ + t ) m ⊕ ( s + ⋯ + s ) m = tm ⊕ sm = ( t + s ) m = 1 ( t + s ) m It results that ∑ i = 1 k t i m ⊕ ∑ i = 1 w s i m ∈ I m .

(I₄) Let a ∈ A and ∑ i = 1 k t i m ∈ I m . Then there exists n ≥ 0 and t ∈ A such that ∑ i = 1 k t i m ≤ n ( tm ) . Hence a ( ∑ i = 1 k t i m ) ≤ a ( tm ⊕ ⋯ ⊕ tm ︸ n times ) = a ( tm + ⋯ + tm ︸ n times ) = ( a ( tm ) + ⋯ + a ( tm ) ︸ n times ) = ( a . t ) m + ⋯ + ( a . t ) m = n ( a . t ) m . It results that a ( ∑ i = 1 k t i m ∈ I m . □

Theorem 4.2. Let V be an A-semimodule, and W be a proper A-ideal of V. Then W is a prime A-ideal of V if and only if for every ideal I of A and A-ideal D of V, ID ⊆ W implies that I ⊆ (W : V) or D ⊆ W.

Proof. (⇒) Let W be a prime A-ideal of V, I be an ideal of A and D be an A-ideal of V such that ID ⊆ W. We show that I ⊆ (W : V) or D ⊆ W. Let I ⊈ (W : V) and D ⊈ W. Then there exist x ∈ I and d ∈ D such that xV ⊈ W and d ∉ W. On the other hand, ID ⊆ W implies that xd ∈ W. Since W is a prime A-ideal of V and d ∉ W, we have xV ⊆ W, which is a contradiction.

(⇐) Let for every ideal I of A and A-ideal D of V, ID ⊆ W implies that I ⊆ (W : V) or D ⊆ W. Let there exist x ∈ A and m ∈ V such that xm ∈ W and m ∉ W. By Lemma 3, let I =≺ x ≻ and D = I _m . Then y ≤ x, for every y ∈ I and so ym ≤ xm ∈ W. Hence, ym ∈ W and so ID = { y ( ∑ i = 1 k t i m ) : y , t ∈ A } = { ∑ i = 1 k t i ( ym ) : y , t ∈ A } ⊆ W . Then I ⊆ (W : V) or D ⊆ W. Since m ∉ W, we have I ⊆ (W : V) and so xV ⊆ W. Therefore, W is a prime A-ideal of V. □

Lemma 4.3. Let V be an A-semimodule and ∅ ≠ W ⊆ V. Then ≺ W ≻ = { w ∈ V : w ≤ w 1 ⊕ ⋯ ⊕ w n ⊕ a 1 r 1 ⊕ ⋯ ⊕ a m r m , for some w 1 , ⋯ , w n , r 1 , ⋯ , r m ∈ W and a 1 , ⋯ , a m ∈ A } is an A-ideal of V that we name it the A-ideal generated by W. In particular, for every α ∈ V, ≺ α ≻ = { x ∈ V : x ≤ n α ⊕ m ( a α ) , ∃ n , m ∈ ℕ and a ∈ A } .

Proof. Let N = { w ∈ V : w ≤ w 1 ⊕ ⋯ ⊕ w n ⊕ a 1 r 1 ⊕ ⋯ ⊕ a m r m , for some w 1 , ⋯ , w n , r 1 , ⋯ , r m ∈ W and a 1 , ⋯ , a m ∈ A } Then we show that N is the smallest A-ideal of V containing W. Let w ∈ W. Then w ≤ w ⊕ 0 and so w ∈ N. It means that W ⊆ N. Let α ≤ β and β ∈ N. Then there exist w₁, ⋯ , w _n , r₁, ⋯ , r _m  ∈ W and a₁, ⋯ , a _m  ∈ A, such that α ≤ β ≤ w 1 ⊕ ⋯ ⊕ w n ⊕ a 1 r 1 ⊕ ⋯ ⊕ a m r m and so α ∈ N. Now, let α, β ∈ N. Then α ≤ w 1 ⊕ ⋯ ⊕ w n ⊕ a 1 r 1 ⊕ ⋯ ⊕ a m r m for some w₁, ⋯ , w _n , r₁, ⋯ , r _m  ∈ W, a₁, ⋯ , a _m  ∈ A and β ≤ t 1 ⊕ ⋯ ⊕ t k ⊕ b 1 z 1 ⊕ ⋯ ⊕ b s z s for some t₁, ⋯ , t _k , z₁, ⋯ , z _s  ∈ W and b₁, ⋯ , b _m  ∈ A. Hence α ⊕ β ≤ w 1 ⊕ ⋯ ⊕ w n ⊕ a 1 r 1 ⊕ ⋯ ⊕ a m r m ⊕ t 1 ⊕ ⋯ ⊕ t k ⊕ b 1 z 1 ⊕ ⋯ ⊕ b s z s and so α ⊕ β ∈ N. Let α ∈ A and w ∈ N. Then w ≤ w 1 ⊕ ⋯ ⊕ w n ⊕ a 1 r 1 ⊕ ⋯ ⊕ a m r m , for some w₁, ⋯ , w _n , r₁, ⋯ , r _m  ∈ W and

a₁, ⋯ , a _m  ∈ A and so α w ≤ α w 1 ⊕ ⋯ ⊕ α w n ⊕ ( α . a 1 ) r 1 ⊕ ⋯ ⊕ ( α . a m ) r m , for some w₁, ⋯ , w _n , r₁, ⋯ , r _m  ∈ W and

α . a₁, ⋯ , α . a _m  ∈ A . It results that αw ∈ N. Hence N is an A-ideal of V containing W. Let K be an A-ideal of V containing W. If α be arbitrary element of K, then α ≤ w 1 ⊕ ⋯ ⊕ w n ⊕ a 1 r 1 ⊕ ⋯ ⊕ a m r m for some w₁, ⋯ , w _n , r₁, ⋯ , r _m  ∈ K, a₁, ⋯ , a _m  ∈ A. Since w 1 ⊕ ⋯ ⊕ w n ⊕ a 1 r 1 ⊕ ⋯ ⊕ a m r m ∈ K we have α ∈ K and so W ⊆ K. Therefore, ≺W ≻ = N. It is clear that, for every α ∈ V, ≺ α ≻ = { x ∈ V : x ≤ n α ⊕ m ( a α ) , ∃ n , m ∈ ℕ and a ∈ A } . □

Theorem 4.4. Let W be a proper A-ideal of V. Then (W : V) = (W : ≺ m ≻), for every m ∈ V \ W if and only if W = {m ∈ V : Jm ⊆ W}, for every ideal J of A such that J ⊈ (W : V).

Proof. (⇒) Let (W : V) = (W : ≺ m ≻), for every m ∈ V \ W and there exists ideal J of A such that J ⊈ (W : V) and W ≠ { m ∈ V : Jm ⊆ W } = L . Then there is m ∈ L such that m ∉ W and so (W : V) = (W : ≺ m ≻). On the other hand, since J ⊈ (W : V), there exists j ∈ J such that jg ∉ W, for some g ∈ V. It results that g ∉ W and so ( W : ≺ g ≻ ) = ( W : V ) = ( W : ≺ m ≻ ) . We prove that (W : ≺ m ≻) ≠ (W : ≺ g ≻). Since m ∈ L, we have jm ∈ W. Now, let x∈ ≺ m ≻. Then x ≤ tm ⊕ k (αm), for some integers t, k ≥ 0 and α ∈ A and so jx ≤ j ( tm ) ⊕ jk ( α m ) = t ( jm ) ⊕ k α ( jm ) ∈ W . It results that jx ∈ W, for every x∈ ≺ m ≻ and so j ∈ (W : ≺ m ≻). But, we have jg ∉ W and so j ∉ (W : ≺ g ≻). Hence (W : ≺ m ≻) ≠ (W : ≺ g ≻), which is a contradiction. Therefore, W = {m ∈ V : Jm ⊆ W}, for every ideal J of A such that J ⊈ (W : V).

(⇐) Let W = {m ∈ V : Jm ⊆ W}, for every ideal J of A such that J ⊈ (W : V). If m ∈ V \ W, then the first we prove that (W : ≺ m ≻) ⊆ (W : V). Let m ∈ V \ W and r ∈ (W : ≺ m ≻). If r ∉ (W : V), then we consider J =≺ r ≻. Thus J ⊈ (W : V) and Jm ⊆ W and so m ∈ V, which is a contradiction. Hence r ∈ (W : V) and so (W : ≺ m ≻) ⊆ (W : V). Now, let r ∈ (W : V). Then rV ⊆ W and so rm ∈ W, for every m ∈ V. We prove that r ∈ (W : ≺ m ≻). Let x∈ ≺ m ≻. Then x ≤ tm ⊕ k (αm), for some integers t, k ≥ 0 and α ∈ A and so rx ≤ r ( tm ) ⊕ rk ( α m ) = t ( rm ) ⊕ k α ( rm ) ∈ W . It results that rx ∈ W, for every x∈ ≺ m ≻ and so r ∈ (W : ≺ m ≻). Therefore, (W : V) ⊆ (W : ≺ m ≻) and therefore, (W : V) = (W : ≺ m ≻).

Proposition 4.5. P = {0} is a prime A-ideal of V if and only if Ann _A  (V) = Z _A  (V), where Z A ( V ) = { r ∈ A : rm = 0 , for some m ∈ V - { 0 } } .

Proof. (⇒) Let P = {0} be a prime A-ideal of V. If a ∈ Ann _A  (V), then am = 0, for every m ∈ V. It results that a ∈ Z _A  (V). Now, let a ∈ Z _A  (V). Then am = 0 ∈ P, for some 0 ≠ m ∈ V. Since m ≠ 0, we have a ∈ (P : V) and so aV = 0. It means that a ∈ Ann _A  (V).

(⇐) Let Ann _A  (V) = Z _A  (V) and am = 0, for a ∈ A and m ∈ V. If m ≠ 0, then a ∈ Z _A  (V) = Ann _A  (V). It results that a ∈ (P : V). □

Theorem 4.6. Let V be a unitary A-semimodule and W be a proper A-ideal of V such that a^◊m ∈ W implies am ∈ W, for every a ∈ A and m ∈ V. If we set P = (W : V), then the following are equivalent:

(a) W is a prime A-ideal of V.

(b) V W is a torsion free A P -semimodule.

(c) W = {m ∈ V : rm ∈ W}, for every r ∈ A \ P.

(d) W = {m ∈ V : Jm ⊆ W}, for every ideal J of A such that J ⊈ P.

(e) P = (W : ≺ m ≻), for all m ∈ V \ W.

(f) P = (W : L), for all A-ideal of V that W ⊈ L.

(g) Ann A ( m W ) = P , for all m ∈ V \ W.

(h) Z A ( V W ) = P .

Proof. (a) ⇒ (b) By Theorem 3, the proof is clear.

(b) ⇒ (c) Let T = {m ∈ V : rm ∈ W}, for every r ∈ A \ P. Consider m ∈ T. Then rm ∈ W and so rm = f _d  (rm, 0) ∈ W. It results that r P m W = rm W = 0 W . Since V W is a torsion free A P -semimodule, we have m W = 0 W and so m ∈ W. Hence W = T.

(c) ⇒ (d) Consider J as an ideal of A such that J ⊈ P. Then there is j ∈ J \ P. Now, let m ∈ {m ∈ V : Jm ⊆ W}. Then jm ∈ W and j ∉ P and so by (c), we have m ∈ W. Hence W = {m ∈ V : Jm ⊆ W}.

(d) ⇒ (e) By Theorem 4, the proof is clear.

(e) ⇒ (f) Let W ⊈ L ⊆ V. Then there is m ∈ L \ W and so by (e), we have P = (W : ≺ m ≻). Since m ∈ L and P = (W : V), we have (W : L) = P.

(f) ⇒ (g) Consider m ∈ V \ W. Let r ∈ Ann A ( m W ) . Then r m W = 0 W and so rm ∈ W. We consider L =≺ m ≻ and so by (f), (W : L) = P. We show that rL ⊆ W. Let x ∈ L. Then x ≤ tm ⊕ k (αm), for some integers t, k ≥ 0 and α ∈ A and so rx ≤ r ( tm ) ⊕ rk ( α m ) = t ( rm ) ⊕ k α ( rm ) ∈ W . It results that rx ∈ W and so r ∈ (L : W) = P. Hence Ann A ( m W ) ⊆ P . We can easily to see that P ⊆ Ann A ( m W ) and so Ann A ( m W ) = P .

(g) ⇒ (h) We have Z A ( V W ) = { r ∈ A : r m W = 0 W , ∃ m ∈ V W \ { 0 W } } = { r ∈ A : f d ( rm , 0 ) ∈ W , ∃ m ∈ V \ W } = { r ∈ A : rm ∈ W , ∃ m ∈ V \ W } = Ann A ( m W ) = P (h) ⇒ (a) The proof is routine. □

5 Conclusion

Since 2003 that Di Nola introduced the definition of MV-modules, many papers have been published in this field. Some researchers tried to present the best definition of modules in algebraic structures. In this paper, we have tried to present a new definition of MV-modules (MV-semimodules), and by this definition, we have investigated some results on Q-ideals of MV-semirings and A-ideals of MV-semimodules. We intend to study MV-semimodules in specific cases, too. For examples, free MV-semimodules, projective(injective) MV-semimodules, and so on. We hope to take an effective steps in this regard.