# On Neutrosophic Soft Prime Ideal.

1 IntroductionBecause of the insufficiency in the available information situation, evaluation of membership values and nonmembership values are not always possible to handle the uncertainties appearing in daily life situations. So there exists an indeterministic part upon which hesitation survives. The neutrosophic set theory by Smarandache [1,2] which is a generalisation of fuzzy set and intuitionistic fuzzy set theory, makes description of the objective world more realistic, practical and very promising in nature. The neutrosophic logic includes the information about the percentage of truth, indeterminacy and falsity grade in several real world problems in law, medicine, engineering, management, industrial, IT sector etc which are not available in intuitionistic fuzzy set theory. But each of the theories suffers from inherent difficulties because of the inadequacy of parametrization tools. Molodtsov [3] introduced a nice concept of soft set theory which is free from the parametrization inadequacy syndrome of different theories dealing with uncertainty. The parametrization tool of soft set theory makes it very convenient and easy to apply in practice. The classical algebraic structures were extended over fuzzy set, intuitionistic fuzzy set, soft set, fuzzy soft set and intuitionistic fuzzy soft set by so many authors, for instance, Rosenfeld [4], Malik and Mordeson [5,6], Lavanya and Kumar [8], Bakhadach et al. [9], Dutta et al. [10- 12], Maji et al. [13], Aktas and Cagman [14], Augunoglu and Aygun [15], Zhang [16], Maheswari and Meera [17] and others.

The notion of neutrosophic soft set theory (NSS) has been innovated by Maji [18]. Later, it has been modified by Deli and Broumi [19]. Cetkin et al. [20,21], Bera and Mahapatra [22-26] and others have produced their research works on fundamental algebraic structures on the NSS theory context.

This paper presents the notion of neutrosophic soft completely prime ideals, neutrosophic soft completely semi-prime ideals and neutrosophic soft prime k-ideals along with investigation of some related properties and theorems. The content of the present paper is designed as following:

Section 2 gives some preliminary useful definitions related to it. In Section 3, neutrosophic soft completely prime ideals is defined and illustrated by suitable examples along with investigation of its structural characteristics. Section 4 deals with the notion of neutrosophic soft completely semi-prime ideals with development of related theorems. The concept of neutrosophic soft prime k-ideals along with some properties has been introduced in Section 6. Finally, the conclusion of our work has been stated in Section 7.

2 Preliminaries

We recall some basic definitions related to fuzzy set, soft set, neutrosophic soft set for the sake of completeness.

2.1 Definition [24]

1. A binary operation *: [0, 1] x [0, 1] [right arrow] [0, 1] is said to be continuous t - norm if * satisfies the following conditions:

(i) * is commutative and associative.

(ii) * is continuous.

(iii) a * 1 = 1 * a = a, [for all] a [member of] [0, 1].

(iv) a * b [less than or equal to] c * d if a [less than or equal to] c, b [less than or equal to] d with a, b, c, d [member of] [0, 1].

A few examples of continuous t-norm are a * b = ab, a * b = min{a, b}, a * b = max{a + b - 1, 0}.

2. A binary operation [??]: [0, 1] x [0, 1] [right arrow] [0, 1] is said to be continuous t-conorm (s-norm) if [??] satisfies the following conditions:

(i) [??] is commutative and associative.

(ii) [??] is continuous.

(iii) a [??] 0 = 0 [??] a = a, [for all]a [member of] [0, 1].

(iv) a [??] b [less than or equal to] c [??] d if a [less than or equal to] c, b [less than or equal to] d with a, b, c, d [member of] [0, 1].

A few examples of continuous s-norm are a [??] b = a + b - ab, a [??] b = max{a, b}, a [??] b = min{a + b, 1}.

2.2 Definition [1]

Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A in X is characterized by a truth-membership function [T.sub.A], an indeterminacy-membership function I a and a falsity-membership function [F.sub.A]. [T.sub.A](x), [I.sub.A](x) and [F.sub.A](x) are real standard or non-standard subsets of ][sup.-]0, [1.sup.+][. That is [T.sub.A], [I.sub.A], [F.sub.A] : X [right arrow] ][sup.-]0, [1.sup.+][. There is no restriction on the sum of [T.sub.A](x), [I.sub.A](x), [F.sub.A](x) and so, [sup.-]0 [less than or equal to] sup [T.sub.A](x) + sup [I.sub.A](x) + sup [F.sub.A](x) [less than or equal to] [3.sup.+].

2.3 Definition [3]

Let U be an initial universe set and E be a set of parameters. Let P(U) denote the power set of U. Then for A [subset or equal to] E, a pair (F, A) is called a soft set over U, where F : A [right arrow] P(U) is a mapping.

2.4 Definition [18]

Let U be an initial universe set and E be a set of parameters. Let NS(U) denote the set of all NSs of U. Then for A [subset or equal to] E, a pair (F, A) is called an NSS over U, where F : A [right arrow] NS(U) is a mapping.

This concept has been redefined by Deli and Broumi [19] as given below.

2.5 Definition [19]

1. Let U be an initial universe set and E be a set of parameters. Let NS(U) denote the set of all NSs of U. Then, a neutrosophic soft set N over U is a set defined by a set valued function [f.sub.N] representing a mapping [f.sub.N] : E [right arrow] NS(U) where [f.sub.N] is called approximate function of the neutrosophic soft set N. In other words, the neutrosophic soft set is a parameterized family of some elements of the set NS(U) and therefore it can be written as a set of ordered pairs,

[mathematical expression not reproducible]

where [mathematical expression not reproducible], respectively called the truth- membership, indeterminacy-membership, falsity-membership function of [f.sub.N](e). Since supremum of each T, I, F is 1 so the inequality [mathematical expression not reproducible] is obvious.

2. Let [N.sub.1] and [N.sub.2] be two NSSs over the common universe (U, E). Then [N.sub.1] is said to be the neutrosophic soft subset of [N.sub.2] if [mathematical expression not reproducible].

We write [N.sub.1] [subset or equal to] [N.sub.2] and then [N.sub.2] is the neutrosophic soft superset of [N.sup.1].

2.6 Proposition [22]

An NSS N over the group (G, o) is called a neutrosophic soft group iff followings hold on the assumption that a * b = minja, b} and a [??] b = max {a, b}.

[mathematical expression not reproducible].

2.7 Definition [24]

1. A neutrosophic soft ring N over the ring (R, +, *) is called a neutrosophic soft left ideal over R if [f.sub.N] (e) is a neutrosophic left ideal of R for each e [member of] E i.e., (i) [f.sub.N] (e) is a neutrosophic subgroup of (R, +) for each e [member of] E and

[mathematical expression not reproducible].

2. A neutrosophic soft ring N over the ring (R, +, *) is called a neutrosophic soft right ideal over R if [f.sub.N](e) is a neutrosophic right ideal of R for each e [member of]1 E i.e., (i) [f.sub.N] (e) is a neutrosophic subgroup of (R, +) for each e [member of] E and

[mathematical expression not reproducible].

3. A neutrosophic soft ring N over the ring (R, +, *) is called a neutrosophic soft ideal over R if [f.sub.N] (e) is a both neutrosophic left and right ideal of R for each e [member of] E.

2.8 Definition [25]

1. Let [phi] : U [right arrow] V and [phi] : E [right arrow] E be two functions where E is the parameter set for each of the crisp sets U and V. Then the pair ([phi], [psi]) is called an NSS function from (U, E) to (V, E). We write, ([phi], [psi]) : (U, E) [right arrow] (V, E). If M is an NSS over U via parametric set E, we shall write (M, E) an NSS over U.

2. Let (M, E), (N, E) be two NSSs defined over U, V respectively and ([phi], [psi]) be an NSS function from (U, E) to (V, E). Then, (i) The image of (M, E) under ([phi], [psi]), denoted by ([phi], [psi]) (M, E), is an NSS over V and is defined by:

([phi], [psi]) (M, E) = ([phi](M), [psi](E)) = {<[phi](a), [f.sub.[phi]](M) >: a [member of] E} where [for all]b [member of] [psi](E), [for all]y [member of] V,

[mathematical expression not reproducible].

(ii) The pre-image of (N, E) under ([phi], [psi]), denoted by [([phi], [psi]).sup.-1] (N, E), is an NSS over U and is defined by :

[([phi], [psi]).sup.-1] (N, E) = ([[phi].sup.-1](N), [[psi].sup.-1](E)) where [for all]a [member of] [[phi].sup.-1] (E), [for all]x [member of] U,

[mathematical expression not reproducible]

If [psi] and [phi] is injective (surjective), then ([phi], [psi]) is injective (surjective).

2.9 Definition [26]

1. An NSS M over (R, E) is said to be constant if each [f.sub.M](e) is constant for e [member of] E i.e., [mathematical expression not reproducible] is same [for all] [member of] E, [for all]x [member of] R.

For M to be nonconstant, if for each e [member of] E the triplet [mathematical expression not reproducible] is at least of two different kinds [for all]x [member of] R.

2. Let R be a ring and M, N be two NSSs over (R, E). Then MoN = L (say) is also an NSS over (R, E) and is defined as following, for e [member of] E and x [member of] R,

[mathematical expression not reproducible].

3. A neutrosophic soft ideal P over (R, E) is said to be a neutrosophic soft prime ideal if (i) P is not constant neutrosophic soft ideal, (ii) for any two neutrosophic soft ideals M, N over (R, E), MoN [subset or equal to] P [??] either M [subset or equal to] P or N [subset or equal to] P.

2.10 Theorem [26]

1. Let P be an NSS over (R, E) such that cardinality of [f.sub.P](e) is 2 i.e., [absolute value of [f.sub.P](e)] = 2 and [absolute value of [f.sub.P](e)]([0.sub.r]) = (1, 0, 0) for each e [member of] E. If [P.sub.0] = {x [member of] R: [[f.sub.p](e)](x) = [[f.sub.p](e)]([0.sub.r])} is a prime ideal over R, then P is a neutrosophic soft prime ideal over (R, E).

2. Let P be an NSS over (R, E). Then P is a neutrosophic soft left (right) ideal over (R, E) iff [??] = {x [member of] R : [[f.sub.P](e)](x) = (1, 0, 0)} with [0.sub.r] [member of] [??] is a left (right) ideal of R.

3. S([not equal to] [phi]) [subset] R is an ideal of R iff there exists a neutrosophic soft ideal M over (R, E) where [f.sub.M]: E [right arrow] NS(R) is defined as, [for all]e [member of] E,

[mathematical expression not reproducible].

with [r.sub.1] > [t.sub.1], [r.sub.2] < [t.sub.2], [r.sub.3] < [t.sub.3] and [r.sub.1], [r.sub.2], [r.sub.3], [t.sub.1], [t.sub.2], [t.sub.3] [member of] [0, 1].

In particular, S([not equal to] [phi]) [subset] R is an ideal of R iff the characteristic function [[chi].sub.S] is a neutrosophic soft ideal over (R, E) where [[chi].sub.S] : E [right arrow] NS(R) is defined as, [for all]e [member of] E,

[mathematical expression not reproducible].

4. An NSS M over (R, E) is a neutrosophic soft left (right) ideal iff each nonempty level set [[[f.sub.M](e)].sub.([alpha], [beta], [gamma])] of the neutrosophic set [f.sub.M](e) is a left (right) ideal of R where [mathematical expression not reproducible].

5. Let P be a neutrosophic soft left (right) ideal over (R, E). Then [P.sub.0] = {x [member of] R : [[f.sub.P](e)](x) = [[f.sub.P](e)]([0.sub.r])} is a left (right) ideal of R.

6. Let P be a neutrosophic soft prime ideal over (R, E). Then [P.sub.0] = {x [member of] R : [[f.sub.P](e)](x) = [[f.sub.P](e)]([0.sub.r])} is a prime ideal of R.

2.11 Definition [7]

A left k-ideal I of a semiring S is a left ideal such that if a [member of] I and x [member of] S and if either a + x [member of] I or x + a [member of] I, then x [member of] I.

Right k-ideal of a semiring is defined dually. A non-empty subset I of a semiring S is called a k-ideal if it is both a left k-ideal and a right k-ideal.

3 Neutrosophic soft completely prime ideal

Here first we have defined a completely prime ideal of a ring and then defined a neutrosophic soft completely prime ideal. These are illustrated with suitable examples. Along with several related properties and theorems have been developed.

Through out this paper, unless otherwise stated, E is treated as the parametric set and e [member of] E, an arbitrary parameter. Moreover the standard t-norm and s- norm are taken into consideration wherever needed through out this paper i.e., a * b = min{a, b} and a [??] b = max{a, b}.

3.1 Definition

An ideal S of a ring R is called a completely prime ideal of R if for x, y [member of] R, xy [member of] S [??] either x [member of] S or y [member of] S.

3.1.1 Example

1. For the ring (Z, +, *) (Z being the set of integers), an ideal (2Z, +, *) is a completely prime ideal.

2. We assume a ring R = {0, x, y, z}. The two binary operations addition and multiplication on R are given by the following tables:

Table 1 + 0 x y z 0 0 x y z x x 0 z y y y z 0 x z z y x 0 Table 2 . 0 x y z 0 0 0 0 0 x 0 0 0 0 y 0 0 y y z 0 0 y y

It is an abelian ring. With respect to these two tables, {0, x} and {0, y} are two ideals of R. From 2nd table, it is evident that {0, x} is a completely prime ideal of R but {0, y} is not so because z x z = y though z E {0, y}.

3. Consider the another ring R = {0, x, y, z} with two binary operations addition and multiplication on R are given by the following tables:

Table 3 + 0 x y z 0 0 x y z x x 0 z y y y z 0 x z z y x 0 Table 4 . 0 x y z 0 0 0 0 0 x 0 0 0 0 y 0 0 0 0 z 0 x y x

It is not an abelian ring. With respect to these two tables, {0, x} is an ideal of R but not completely prime ideal. Because y * z = 0, z * z = x, y * y = 0 but y, z [not member of] {0, x}.

3.2 Proposition

If S is a completely prime ideal of a ring R then S is a prime ideal of R.

Proof. Let S be a completely prime ideal of a ring R and A, B be two ideals of R such that AB [subset or equal to] S. Suppose A [subset not equal to] S and B [subset not equal to] S. Then there exists x [member of] A and y [member of] B such that x, y [not member of] S. But xy [member of] S as AB [subset or equal to] S. Since S is a completely prime ideal of R, so either x [member of] S or y [member of] S and this leads a contradiction to the fact x, y [not member of] S. Hence S is a prime ideal of R.

3.3 Definition

A neutrosophic soft ideal N over (R, E) is called a neutrosophic soft completely prime ideal if [for all]x, y [member of] R and [for all]e [member of] E,

[mathematical expression not reproducible].

3.3.1 Example

Consider the Example [3.1.1](2). We define an NSS M over (R, E) as following, [for all]r [member of] R and [for all]e [member of] E,

[mathematical expression not reproducible].

Then M is a neutrosophic soft completely prime ideal over (R, E).

3.4 Theorem

An NSS N is a neutrosophic soft completely prime ideal over (R, E) iff for e [member of] E, [absolute value of [f.sub.N](e)] = 2, [[f.sub.N](e)]([0.sub.r]) = (1, 0, 0) and [??] = {x [member of] R : [[f.sub.N](e)](x) = (1, 0, 0)} is a completely prime ideal of R.

Proof. Let N be a neutrosophic soft completely prime ideal over (R, E). Then N is a neutrosophic soft ideal over (R, E) and so [??] is an ideal over R by Theorem [2.11] (2). To prove N is a complete prime ideal, let xy [member of] [??] for x, y [member of] R. Then [[f.sub.N](e)](xy) = (1, 0, 0) for e [member of] E. But,

[mathematical expression not reproducible];

This implies that

[mathematical expression not reproducible];

This shows that,

[mathematical expression not reproducible];

But [mathematical expression not reproducible]. Hense [mathematical expression not reproducible]. Thus [??] is a complete prime ideal.

Conversely suppose [??] is a completely prime ideal with the given conditions. As [??] is an ideal of R, so N is a neutrosophic soft ideal over (R, E) by Theorem [2.11] (2). For contrary, suppose N is not neutrosophic soft completely prime ideal. Then,

[mathematical expression not reproducible];

Since [absolute value of [f.sub.N](e)] = 2 and [[f.sub.N](e)]([0.sub.r]) = (1, 0, 0) then there exists x, y [member of] R so that [[f.sub.N](e)](x) = [[f.sub.N](e)](y) = ([r.sub.1], [r.sub.2], [r.sub.3]) [not equal to] (1, 0, 0) (say) for 0 [less than or equal to] [r.sub.1] < 1 and 0 < [r.sub.2], [r.sub.3] [less than or equal to] 1. Then,

[mathematical expression not reproducible];

Since [??] is completely prime ideal, so either x [member of] [??] or y [member of] [??] i.e., [[f.sub.N](e)](x) = [[f.sub.N](e)](y) = (1, 0, 0). A contradiction arises to the fact that [[f.sub.N](e)](x) = [[f.sub.N](e)](y) = ([r.sub.1], [r.sub.2], [r.sub.3]) [not equal to] (1, 0, 0). Thus,

[mathematical expression not reproducible];

and so N is a neutrosophic soft completely prime ideal over (R, E).

3.5 Theorem

Let N be a neutrosophic soft completely prime ideal over (R, E) with [absolute value of [f.sub.N](e)] = 2, [[f.sub.N](e)]([0.sub.r]) = (1, 0, 0) for each e [member of] E. Then N is a neutrosophic soft prime ideal over (R, E).

Proof. Let the condition hold. By Theorem [3.4], [??] = {x [member of] R : [[f.sub.N](e)](x) = (1, 0, 0)} is a completely prime ideal of R. Then by Proposition [3.2], [??] is a prime ideal of R. Hence N is a neutrosophic soft prime ideal over (R, E) by Theorem [2.11](1).

3.6 Theorem

Let R be a ring. Then S([not equal to] [phi]) [subset] R be a completely prime ideal of R iff an NSS N over (R, E) is a neutrosophic soft completely prime ideal where [f.sub.N] : E [right arrow] NS(R) is defined as:

[mathematical expression not reproducible].

with [r.sub.1] > [t.sub.1], [r.sub.2] < [t.sub.2], [r.sub.3] < [t.sub.3] and [r.sub.1], [r.sub.2], [r.sub.3], [t.sub.1], [t.sub.2], [t.sub.3] [member of] [0, 1].

Proof. First let S([not equal to] [phi]) [subset] R be a completely prime ideal of R. Then S is an ideal of R and so by Theorem [2.11](3), N is a neutrosophic soft ideal over (R, E). To end the theorem, we shall just show that N is completely prime. For contrary, suppose

[mathematical expression not reproducible];

Then by definition of [f.sub.N](e), we have [[f.sub.N](e)](xy) = ([r.sub.1], [r.sub.2], [r.sub.3]) and [[f.sub.N](e)](x) = [[f.sub.N](e)](y) = ([t.sub.1], [t.sub.2], [t.sub.3]). This implies xy [member of] S but x, y [member of] S which is a contradiction to the fact that S is a completely prime ideal of R. Hence N is a neutrosophic soft completely prime ideal over (R, E).

Conversely, let N in given form be a neutrosophic soft completely prime ideal over (R, E). Then N is a neutrosophic soft ideal over (R, E) and so by Theorem [2.11](3), S is an ideal of R. To show S is a completely prime ideal of R, let xy [member of] S. Then,

[mathematical expression not reproducible]

Thus S is a completely prime ideal of R.

3.6.1 Corollary

A non empty subset S of a ring R is a completely prime ideal iff the characteristic function \s is a neutrosophic soft completely prime ideal over (R, E) where \s : E [right arrow] NS(R) is defined by :

[mathematical expression not reproducible].

Proof. It is the particular case of Theorem [3.6].

3.7 Theorem

An NSS M over (R, E) is a neutrosophic soft completely prime ideal means each nonempty level set [[[f.sub.M](e)].sub.([alpha],[beta],[gamma])] of the neutrosophic set [f.sub.M](e), e [member of] E is a completely prime ideal of R where [mathematical expression not reproducible].

Proof. Here M is a neutrosophic soft completely prime ideal over (R, E). Then M is a neutrosophic soft ideal over (R,E) and so by Theorem [2.11](4), [[[f.sub.M](e)].sub.([alpha],[beta],[gamma])] is an ideal of R. To complete the theorem, let xy [member of] [[[f.sub.M](e)].sub.([alpha],[beta],[gamma])]. Then,

[mathematical expression not reproducible]

Thus [[[f.sub.M](e)].sub.([alpha],[beta],[gamma])] is a completely prime ideal of R.

3.8 Proposition

Let S be a completely prime ideal of a ring R. Then there exists a neutrosophic soft completely prime ideal M over (R, E) such that [[[f.sub.M](e)].sub.([alpha],[beta],[gamma])] = S for e [member of] E and [alpha], [beta], [gamma] [member of] (0, 1).

Proof. As S is a completely prime ideal of a ring R, so S is an ideal of R. For [alpha], [beta], [gamma] [member of] (0, 1) define an NSS M over (R, E) as following :

[mathematical expression not reproducible].

Then by Theorem [2.11](3), M is a neutrosophic soft ideal over (R, E). If possible let M is not a neutrosophic soft completely prime ideal over (R, E). Then,

[mathematical expression not reproducible];

Then by definition of [f.sub.M](e), we have [[f.sub.M](e)](xy) = ([alpha], [beta], [gamma]) and [[f.sub.M](e)](x) = [[f.sub.M](e)](y) = (0, 1, 1). This implies xy [member of] S but x, y [member of] S which is a contradiction to the fact that S is a completely prime ideal of R. Hence M is a neutrosophic soft completely prime ideal over (R, E). Obviously [[[f.sub.M](e)].sub.([alpha],[beta],[gamma])] = S for each e E E.

3.9 Theorem

Let ([phi], [psi]) : ([R.sub.1], E) [right arrow] ([R.sub.2], E) be a neutrosophic soft homomorphism where [R.sub.1], [R.sub.2] be two rings. Suppose (M, E) and (N, E) be two neutrosophic soft left (right) ideals over [R.sub.1] and [R.sub.2], respectively. Then,

1. ([phi], [psi])(M, E) is a neutrosophic soft left (right) ideal over [R.sub.2] if ([phi], [psi]) is epimorphism.

2. [([phi], [psi]).sup.-1](N, E) is a neutrosophic soft left (right) ideal over [R.sub.1].

Proof. 1. Let b [member of] [phi](E) and [y.sub.1], [y.sub.2], s [member of] [R.sub.2]. For [[phi].sup.-1]([y.sub.1]) = [phi] or [[phi].sup.-1]([y.sub.2]) = [phi], the proof is straight forward.

So, we assume that there exists [x.sub.1], [x.sub.2], r [member of] [R.sub.1] such that [phi]([x.sub.1]) = [y.sub.1], [phi]([x.sub.2]) = [y.sub.2], [phi](r) = s. Then,

[mathematical expression not reproducible]

Since, this inequality is satisfied for each [x.sub.1], [x.sub.2] [member of] [R.sub.1] satisfying [phi]([x.sub.1]) = [y.sub.1], [phi]([x.sub.2]) = [y.sub.2] so we have,

[mathematical expression not reproducible]

Also, [mathematical expression not reproducible] Next,

[mathematical expression not reproducible]

Since, this inequality is satisfied for each [x.sub.1], [x.sub.2] [member of] [R.sub.1] satisfying [phi]([x.sub.1]) = [y.sub.1], [phi]([x.sub.2]) = [y.sub.2] so we have,

[mathematical expression not reproducible]

Also, [mathematical expression not reproducible]. Similarly, we can show that

[mathematical expression not reproducible];

This completes the proof.

2. For a [member of] [[phi].sup.-1][(E) and [x.sub.1], [x.sub.2] [member of] [R.sub.1], we have,

[mathematical expression not reproducible]

Next,

[mathematical expression not reproducible]

Similarly, [mathematical expression not reproducible];

This proves the 2nd part.

3.10 Theorem

Let ([phi], [psi]) be a neutrosophic soft homomorphism from a ring [R.sub.1] to a ring [R.sub.2]. Suppose (M, E) and (N, E) are neutrosophic soft completely prime ideals over [R.sub.1] and [R.sub.2], respectively. Then,

1. ([phi], [psi]) (M, E) is a neutrosophic soft completely prime ideal over [R.sub.2].

2. [([phi], [psi]).sup.-1](N, E) is a neutrosophic soft completely prime ideal over [R.sub.1].

Proof. 1. If possible, let (M, E) be a neutrosophic soft completely prime ideal over [R.sub.1] but ([phi], [psi]) (M, E) is not so over [R.sub.2]. Then for b [member of] [psi](E) and [y.sub.1], [y.sub.2] [member of] [R.sub.2],

[mathematical expression not reproducible]

Since the inequality holds for each [x.sub.1], [x.sub.2] [member of] [R.sub.1] satisfying [phi]([x.sub.1]) = [y.sub.1], [phi]([x.sub.2]) = [y.sub.2] so we have [mathematical expression not reproducible] which is a contradiction to the truth that (M, E) is a neutrosophic soft completely prime ideal over [R.sub.1]. We can reach to the same conclusion taking the indeterminacy membership function (I) and falsity membership function (F) also. Hence we get the first result.

2. For a [member of] [[psi].sup.-1](E) and [x.sub.1], [x.sub.2] [member of] [R.sub.1], we have,

[mathematical expression not reproducible]

This shows the 2nd result.

4 Neutrosophic Soft Completely Semi-Prime Ideal

In this section the concept of semi-prime ideal, completely semi-prime ideal of a ring R and neutrosophic soft completely semi-prime ideal are focussed.

4.1 Definition

1. An ideal I of a ring R is called a semi-prime ideal if there is another ideal J of R such that JJ [subset or equal to] I [??] J [subset or equal to] I.

2. An ideal J of a ring R is called a completely semi-prime ideal if for x [member of] R, xx [member of] J [??] x [member of] J. xx is denoted by [x.sup.2].

4.1.1 Example

1. Let R = {0, x, y, z} be a ring. The two binary operations addition and multiplication on R are given by the following tables :

Table 5 + 0 x y z 0 0 x y z x x 0 z y y y z 0 x z z y x 0 Table 6 . 0 x y z 0 0 0 0 0 x 0 x x 0 y 0 x y z z 0 0 z z

Then {0, x} is a completely semi-prime ideal of R as 0 * 0 = 0, x * x = x, y * y = y, z * z = z.

2. Consider the Example [3.1.1](3). Then {0, x} is not a completely semi-prime ideal, because z * z = x, y * y = 0 but y, z [member of] {0, x}.

4.2 Proposition

Every completely prime ideal of a ring R is a completely semi-prime ideal of R. Proof. By taking y = x, the proof follows directly from Definition [3.1].

4.3 Definition

Let R be a ring and E be a parametric set. A neutrosophic soft ideal N over (R, E) is called a neutrosophic soft completely semi-prime ideal if [for all]x, y [member of] R and [for all]e [member of] E,

[mathematical expression not reproducible].

4.3.1 Example

Consider the Example [4.1.1](1). We define an NSS M over (R,E) as following, [for all]r [member of] R and [for all]e [member of] E,

[mathematical expression not reproducible].

Then M is a neutrosophic soft completely semi-prime ideal over (R, E).

4.4 Lemma

A neutrosophic soft ideal N over (R, E) is a neutrosophic soft completely semi- prime ideal iff [[f.sub.N](e)]([x.sup.2]) = [[f.sub.N](e)](x), for every e [member of] E, x [member of] R.

Proof. Let N be a neutrosophic soft ideal over (R,E) with [[f.sub.N](e)]([x.sub.2]) = [[f.sub.N](e)](x), [for all]e [member of] E and [for all]x [member of] R. Then by Definition [4.3], N is a neutrosophic soft completely semi-prime ideal over (R,E).

Conversely, if N is a neutrosophic soft completely semi-prime ideal by Definition [4.3], [mathematical expression not reproducible] and as N is a neutrosophic soft ideal over (R, E), then [mathematical expression not reproducible].

4.5 Theorem

An NSS N over (R, E) is a neutrosophic soft completely semi-prime ideal iff for e [member of] E, S = {x [member of] R : [[f.sub.N](e)](x) = [[f.sub.N](e)]([0.sub.r])}, [0.sub.r] being the additive identity of ring R, is a completely semi-prime ideal of R.

Proof. Let N be a neutrosophic soft completely semi-prime ideal over (R, E). Then [[f.sub.N](e)]([x.sup.2]) = [[f.sub.N](e)](x) for every e [member of] E, x [member of] R. Now let [x.sup.2] [member of] S. Then [[f.sub.N](e)]([x.sup.2]) = [[f.sub.N](e)]([0.sub.r]) [??] [[f.sub.N](e)](x) = [[f.sub.N](e)]([0.sub.r]) [??] x [member of] S. Hence S is a completely semi-prime ideal of R.

Conversely, if S is a completely semi-prime ideal of R. Then [x.sup.2] [member of] S [right arrow] x [member of] S. Since [x.sup.2] [member of] S, then [[f.sub.N](e)]([x.sup.2]) = [[f.sub.N](e)]([0.sub.r]) and [[f.sub.N](e)](x) = [[f.sub.N](e)]([0.sub.r]) [right arrow] [[f.sub.N](e)]([x.sup.2]) = [[f.sub.N](e)](x). Hence by Lemma [4.4], N is a neutrosophic soft completely semi-prime ideal over (R, E).

4.6 Theorem

An NSS N is a neutrosophic soft completely semi-prime ideal over (R, E) iff [[[f.sub.N](e)].sub.([alpha],[beta],[gamma])] is a completely semi-prime ideal of R where [mathematical expression not reproducible].

Proof. Let N be a neutrosophic soft completely semi-prime ideal over (R, E). Then [[f.sub.N](e)]([x.sup.2]) = [[f.sub.N](e)](x). Now

[mathematical expression not reproducible]

Hence, [[[f.sub.N](e)].sub.([alpha],[beta],[gamma])] is a completely semi-prime ideal of R.

Conversely, let [[[f.sub.N](e)].sub.([alpha],[beta],[gamma])] be a completely semi-prime ideal of R. Then [x.sup.2] [member of] [[[f.sub.N](e)].sub.([alpha],[beta],[gamma])] [??] x [member of] [[[f.sub.N](e)].sub.([alpha],[beta],[gamma])] i.e.,

[mathematical expression not reproducible]

Now, suppose [[f.sub.N](e)]([x.sup.2]) = [[f.sub.N](e)](x). Let [[f.sub.N](e)](x) = ([t.sub.1], [t.sub.2], [t.sub.3]). Then [mathematical expression not reproducible] which is a contradiction as [[[f.sub.N](e)].sub.([alpha],[beta],[gamma])] is a completely semi-prime ideal of R. Hence [[f.sub.N](e)]([x.sup.2]) = [[f.sub.N](e)](x) and so N is a neutrosophic soft completely semi-prime ideal over (R, E) by Lemma [4.4].

4.7 Theorem

Let ([phi], [psi]) be a neutrosophic soft homomorphism from a ring [R.sub.1] to a ring [R.sub.2]. Suppose (M, E) and (N, E) are neutrosophic soft completely semi-prime ideals over [R.sub.1] and [R.sub.2], respectively. Then,

1. ([phi], [psi]) (M, E) is a neutrosophic soft completely semi-prime ideal over [R.sub.2].

2. [([phi], [psi]).sup.-1] (N, E) is a neutrosophic soft completely semi-prime ideal over [R.sub.1].

Proof. 1. If possible, let (M,E) be a neutrosophic soft completely semi-prime ideal over [R.sub.1] but ([phi], [psi]) (M, E) is not so over [R.sub.2]. Then for b [member of] [psi](E) and y [member of] [R.sub.2],

[mathematical expression not reproducible]

Since the inequality holds for each x [member of] [R.sub.1] satisfying [phi](x) = y, so we have [mathematical expression not reproducible] which is a contradiction to the fact that (M, E) is a neutrosophic soft completely semi-prime ideal over [R.sub.1]. We can reach to the same conclusion taking the indeterminacy membership function (I) and falsity membership function (F) also. Hence we get the first result.

2. For a [member of] [[psi].sup.-1](E) and x [member of] [R.sub.1], we have,

[mathematical expression not reproducible];

This proves the 2nd result.

5 Neutrosophic soft prime k-ideal

5.1 Definition

A neutrosophic soft ideal N over (R, E) is said to be a neutrosophic soft k- ideal over (R, E) if [for all]x, y [member of] R and [for all]e [member of] E,

[mathematical expression not reproducible].

5.1.1 Example

1. Let Z be the set of all integers and E = {[e.sub.1], [e.sub.2], [e.sub.3]} be a parametric set. We consider an NSS N over (Z, E) given by the following table :

Table 7 [f.sub.N] [f.sub.N] [f.sub.N] ([e.sub.1]) ([e.sub.2]) ([e.sub.3]) [Z.sub.1] (0.3, 0.8, 0.5) (0.4, 0.5, 0.7) (0.7, 0.6, 0.4) [Z.sub.2] (0.4, 0.6, 0.3) (0.6, 0.2, 0.4) (0.7, 0.4, 0.2) [Z.sub.3] (0.6, 0.2, 0.1) (1, 0, 0) (0.9, 0.1, 0.1)

where [Z.sub.i] = {[+ or -] 1, [+ or -] 3, [+ or -] 5, ...}, [Z.sub.2] = {[+ or -] 2, [+ or -] 4, [+ or -] 6, ...}, [Z.sub.3] = {0}. Then N is a neutrosophic soft k-ideal over (Z, E). To verify it, we shall show

(i) [f.sub.N](e) is neutrosophic subgroup of (Z, +) for each e [member of] E.

(ii) [f.sub.N](e) is both neutrosophic left and right ideal of Z for each e [member of] E.

(iii) [f.sub.N](e) is neutrosophic k-ideal of Z for each e [member of] E.

If x E [Z.sub.1], y E [Z.sub.2] then x - y E [Z.sub.1]. We then write [Z.sub.1] - [Z.sub.2] = [Z.sub.1] and so on. Here [Z.sub.1] - [Z.sub.1] = [Z.sub.2] or [Z.sub.3], [Z.sub.1] - [Z.sub.2] = [Z.sub.1], [Z.sub.1] - [Z.sub.3] = [Z.sub.3], [Z.sub.2] - [Z.sub.2] = [Z.sub.2] or [Z.sub.3], [Z.sub.2] - [Z.sub.3] = [Z.sub.2], [Z.sub.3] - [Z.sub.3] = [Z.sub.3]. Then Table 7 shows the result (i) obviously.

Next [Z.sub.1].[Z.sub.1] = [Z.sub.1], [Z.sub.2].[Z.sub.2] = [Z.sub.2], [Z.sub.3].[Z.sub.3] = [Z.sub.3], [Z.sub.2].[Z.sub.1] = [Z.sub.1].[Z.sub.2] = [Z.sub.2], [Z.sub.1].[Z.sub.3] = [Z.sub.3].[Z.sub.1] = [Z.sub.3], [Z.sub.2].[Z.sub.3] = [Z.sub.3].[Z.sub.2] = [Z.sub.3]. Then the result (ii) also holds by Table 7.

Finally [Z.sub.1] + [Z.sub.1] = [Z.sub.2] or [Z.sub.3], [Z.sub.1] + [Z.sub.2] = [Z.sub.1], [Z.sub.1] + [Z.sub.3] = [Z.sub.3], [Z.sub.2] + [Z.sub.2] = [Z.sub.2] or [Z.sub.3], [Z.sub.2] + [Z.sub.3] = [Z.sub.2], [Z.sub.3] + [Z.sub.3] = [Z.sub.3]. The Table 7 then meets the result (iii) clearly.

2. Let R be the set of real numbers and E = {[e.sub.1], [e.sub.2], [e.sub.3]} be a parametric set. Consider an NSS M over (R, E) given by the following table :

Table 8 [f.sub.M] [f.sub.M] [f.sub.M] ([e.sub.1]) ([e.sub.2]) ([e.sub.3]) Q (0.6, 0.1, 0.3) (0.8, 0.2, 0.4) (0.5, 0.6, 0.7) [Q.sup.c] (0.5, 0.4, 0.7) (0.4, 0.5, 0.6) (0.3, 0.7, 1)

where Q and [Q.sup.c] are the set of rational and irrational numbers, respectively. If x [member of] Q, y [member of] [Q.sup.c] then x - y [member of] [Q.sup.c]. We write Q - [Q.sup.c] = [Q.sup.c] and so on.

Then Q - Q = Q, Q - [Q.sup.c] = [Q.sup.c] [Q.sup.c] - [Q.sup.c] = Q or [Q.sup.c]. Clearly [f.sub.M](e) is neutrosophic subgroup of (R, +) for each e E E by Table 8.

Next, Q.Q = Q, Q.[Q.sup.c] = [Q.sup.c] [Q.sup.c].[Q.sup.c] = Q or [Q.sup.c]. Then Table 8 shows that [f.sub.M](e) is neutrosophic ideal of R for each e E E.

Finally Q + Q = Q, Q + [Q.sup.c] = [Q.sup.c] [Q.sup.c] + [Q.sup.c] = Q or [Q.sup.c]. Then [f.sub.M](e) is neutrosophic k-ideal of R for each e E E by Table 8.

Hence M is a neutrosophic soft k-ideal over (R, E).

5.2 Definition

A neutrosophic soft k-ideal P over (R, E) is said to be a neutrosophic soft prime k-ideal if (i) P is not constant over (R, E), (ii) for any two neutrosophic soft ideals M, N over (r, E), MoN [subset or equal to] P [??] either M [subset or equal to] P or N [subset or equal to] P.

5.3 Theorem

Let P be a neutrosophic soft prime k-ideal over (R, E). Then [P.sub.0] = {x [member of] R : [[f.sub.P](e)](x) = [[f.sub.P](e)]([0.sub.r]), [for all]e [member of] E} is a prime k-ideal of R.

Proof. Let x, x + y [member of] [P.sub.0] for x, y [member of] R. Then [[f.sub.P](e)](x) = [[f.sub.P](e)](x + y) = [[f.sub.P](e)]([0.sub.r]). Since P is a neutrosophic soft k-ideal over (R, E), so [for all]e [member of] E,

[mathematical expression not reproducible];

But [mathematical expression not reproducible]. Hence [P.sub.0] is a k-ideal of R. Also by Theorem [2.11](6), [P.sub.0] is a prime ideal of R. This completes the proof.

5.4 Theorem

Let P be a neutrosophic soft prime k-ideal over (Z, E), Z being the set of integers with [P.sub.0] = {x [member of] R : [[f.sub.P](e)](x) = [[f.sub.P](e)](0), [for all]e [member of] E} = nZ, n being a natural number. Then [absolute value of [f.sub.P](e)] [less than or equal to] r, where r is the number of distinct positive divisor of n.

Proof. Let a([not equal to] 0) be an integer and d = gcd(a, n). Then there exists r, s [member of] Z - {0} such that ns = ar + d or ar = ns + d. We shall now estimate following two cases : Case 1 : When ns = ar + d, then [for all]e [member of] E and as n [member of] [P.sub.0] = nZ,

[mathematical expression not reproducible];

Again P is a neutrosophic soft k-ideal over (Z, E). So,

[mathematical expression not reproducible];

Case 2 : When ar = ns + d, then [for all]e [member of] E and as n [member of] [P.sub.0] = nZ,

[mathematical expression not reproducible];

Again,

[mathematical expression not reproducible];

Now as P is a neutrosophic soft k-ideal over (Z, E) so,

[mathematical expression not reproducible];

Thus in either case [for all]e [member of] E,

[mathematical expression not reproducible];

Further since d is a divisor of a, there exists t [member of] Z - {0} such that a = dt. So [for all]e [member of] E,

[mathematical expression not reproducible];

Hence [mathematical expression not reproducible].

Thus for any integer a([not equal to] 0) there exists a divisor d of n such that [[f.sub.P](e)](d) = [[f.sub.P](e)](a), [for all]e [member of] E.

[mathematical expression not reproducible].

This follows the theorem.

5.5 Lemma

For a neutrosophic soft prime k-ideal N over (Z, E)(Z being the set of integers), [N.sub.0] = pZ is a prime k-ideal of Z iff p is either zero or prime.

This result is similar to the matter incase of prime ideal in the ring of integers in classical sense. So the proof is omitted.

5.6 Theorem

Let N be a neutrosophic soft prime k-ideal over (Z, E), Z being the set of integers. Then [absolute value of [f.sub.N](e)] = 2 for each e [member of] E.

Conversely, if N is an NSS over (Z, E) such that for each e [member of] E, [[f.sub.N](e)](x) = (1, 0, 0) when p|x and [[f.sub.N](e)](x) = ([alpha], [beta], [gamma]) when p [??]x, p being a fixed prime and [beta] > 0, [gamma] > 0, [alpha] < 1, then N be a neutrosophic soft prime k-ideal over (Z, E).

Proof. Let N be a neutrosophic soft prime k-ideal over (Z, E) with [N.sub.0] = pZ. By Theorem [5.3], [N.sub.0] is a prime k-ideal of Z. Hence by Lemma [5.5], p is prime i.e., p has only two distinct divisors namely 1,p. So by Theorem [5.4], [absolute value of [f.sub.N](e)] [less than or equal to] 2. But N being a neutrosophic soft prime k-ideal can not be constant, so [absolute value of [f.sub.N](e)] = 2, [for all]e [member of] E. Conversely, let N be an NSS over (Z, E) satisfying the given conditions. Let x, y [member of] Z.

If [mathematical expression not reproducible] = 1 or [alpha] and so [mathematical expression not reproducible].

If [mathematical expression not reproducible] = then p|x and p|y. It implies p(x + y) and [mathematical expression not reproducible].

Thus in either case [mathematical expression not reproducible].

Next, if [mathematical expression not reproducible] and so, [mathematical expression not reproducible].

If [mathematical expression not reproducible]. It implies p(x + y) and [mathematical expression not reproducible].

Thus in either case [mathematical expression not reproducible].

Finally, if [mathematical expression not reproducible] then [mathematical expression not reproducible] = 0 or [beta] and so [mathematical expression not reproducible].

If [mathematical expression not reproducible] = 0 then p|x and p|y. It implies p|(x + y) and [mathematical expression not reproducible].

Thus in either case [mathematical expression not reproducible].

Further if [mathematical expression not reproducible] then either [mathematical expression not reproducible].

If [mathematical expression not reproducible]. Then [mathematical expression not reproducible]. Thus in either case we have [mathematical expression not reproducible].

So N is a neutrosophic soft ideal over (Z, E).

We shall now prove that N is a neutrosophic soft k-ideal over (Z, E).

If [[f.sub.N](e)](x + y) = ([alpha], [beta], [gamma]) or [[f.sub.N](e)](y) = ([alpha], [beta], [gamma]), then the inequalities in Definition [5.1] are obvious.

If [[f.sub.N](e)](x + y) = (1, 0, 0) or [[f.sub.N](e)](y) = (1, 0, 0), then p|(x + y) and p|y. It implies p|x and so [[f.sub.N](e)](x) = (1, 0, 0). Thus the inequalities in Definition [5.1] hold clearly.

Therefore N is a neutrosophic soft k-ideal over (Z, E) and so [N.sub.0] is a k- ideal over Z.

Finally, we shall prove that N is a neutrosophic soft prime k-ideal over (Z, E).

To prove it, we shall first show that [N.sub.0] = pZ is a prime k-ideal of Z. Now, x [member of] [N.sub.0] [??] [[f.sub.N](e)](x) = [[f.sub.N](e)](0) = (1, 0, 0) [??] p|x [??] x = pm, m [member of] Z [??] x [member of] pZ.

Thus [N.sub.0] = pZ, p being a prime and so [N.sub.0] is a prime k-ideal of Z by Lemma [5.5].

Further, [absolute value of [f.sub.N](e)] = 2, [for all]e [member of] E namely (1, 0, 0) and ([alpha], [beta], [gamma]). So N is not constant over (Z, E). Now assume two neutrosophic soft ideals S, Q over (Z, E) such that SoQ [subset or equal to] N and S [subset not equal to] N, Q [subset not equal to] N. Then there exists x, y [member of] Z such that [mathematical expression not reproducible]. Then [[f.sub.N](e)](x) = [[f.sub.N](e)](y) = ([alpha], [beta], [gamma]) obviously and so x, y [not member of] [N.sub.0]. It implies xy [not member of] [N.sub.0] as it is a prime k-ideal of an abelian ring Z. So [[f.sub.N](e)](xy) = ([alpha], [beta], [gamma]). Thus [mathematical expression not reproducible]. But,

[mathematical expression not reproducible];

It opposes the fact. This ends the theorem.

6 Conclusion

The aim of this paper is to put forward the study of the concept neutrosophic soft prime ideal introduced in [26]. Here we have studied about neutrosophic soft completely prime ideal, neutrosophic soft completely semi-prime ideal and neutrosophic soft prime k-ideal. They are defined and illustrated by suitable examples. Their related properties and structural characteristics have been investigated also. Moreover a number of theorems have been developed in virtue of these notions. The concepts will bring a new opportunity in research and development of algebraic structures over NSS theory context, we expect.

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Received: April 9, 2018. Accepted: April 23, 2018.

Tuhin Bera (1) and Nirmal Kumar Mahapatra (2)

(1) Department of Mathematics, Boror S. S. High School, Bagnan, Howrah-711312, WB, India, E-mail: tuhin78bera@gmail.com

(2) Department of Mathematics, Panskura Banamali College, Panskura RS-721152, WB, India, E-mail: nirmal_hridoy@yahoo.co.in

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Author: | Bera, Tuhin; Mahapatra, Nirmal Kumar |
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Publication: | Neutrosophic Sets and Systems |

Article Type: | Report |

Date: | Jun 1, 2018 |

Words: | 8573 |

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