Proper Fuzziﬁcation of Prime Ideals of a Hemiring

Prime fuzzy ideals, prime fuzzy k -ideals, and prime fuzzy h -ideals are roped in one condition. It is shown that this way better fuzziﬁcation is achieved. Other major results of the paper are: every fuzzy ideal (resp., k -ideal, h -ideal) is contained in a prime fuzzy ideal (resp., k -ideal, h -ideal). Prime radicals and nil radicals of a fuzzy ideal are deﬁned; their relationship is established. The nil radical of a fuzzy k -ideal (resp., an h -ideal) is proved to be a fuzzy k -ideal (resp., h -ideal). The correspondence theorems for di ﬀ erent types of fuzzy ideals of hemirings are established. The concept of primary fuzzy ideal is introduced. Minimum imperative for proper fuzziﬁcation is suggested and it is shown that the fuzziﬁcations introduced in this paper are proper fuzziﬁcations.


Introduction
This paper is, in some sense, an extended version of the article "On Fuzzification of Prime Ideals with Special Reference to Semirings" in SciTopics and something more.
Several attempts have been made to fuzzify the concepts of prime ideals/k-ideals/h-ideals of a semiring [1][2][3][4][5][6][7], prime ideals of a ring [8][9][10][11][12][13][14][15], and prime ideals of a semigroup [16][17][18].We have discussed elsewhere [6], in detail, the deficiencies in the definition of a prime fuzzy h-ideal proposed in [7].The definition suffers from three major drawbacks.First, it is very restrictive in the sense that the fuzzy h-ideals, which are prime according to the definition, are 2-valued function.Secondly, since one of the two values is always 1 (the greatest element of the lattice), the function is determined by only one value, thus, severely curtailing its fuzziness.Third, when the zero element of the valuation lattice is not a prime element (and this happens in many important lattices), even the characteristic function of a prime ideal fails to be a prime fuzzy ideal.The technique adopted for the fuzzification by Zhan and Dudek in [7] and by others in [1][2][3]5] is identical.Therefore, their prime fuzzy ideals inherit the same drawbacks.In [6] we have redefined prime fuzzy left h-ideal so that these deficiencies are completely removed.(It should be thankfully mentioned that one of the referees of the present paper has pointed out that in [4] two similar definitions of prime fuzzy ideal are stated.However, while proving major results of the paper, only 2-valued prime fuzzy ideals are used.)In this paper, we show that the problem of fuzzification of left ideal, left k-ideal, and left h-ideal need not be tackled separately.One single condition governs all the three.We also "refine" our definitions so that they look more compact, elegant, and easy for application.We prove that every proper fuzzy ideal (resp., k-ideal, h-ideal) is contained in a prime fuzzy ideal (resp., k-ideal, h-ideal).We introduce the concepts of fuzzy prime radical (or to be more precise, prime radical of a fuzzy ideal) and fuzzy nil radical (or nil radical of a fuzzy ideal), and fuzzy primary ideal.The prime and the nil radicals of a fuzzy k-ideal coincide when the valuation lattice is linearly ordered (e.g., when it is [0, 1]).An analogous result holds for fuzzy h-ideals.We establish a correspondence between fuzzy ideals (resp., k-ideals, h-ideals) of a hemiring and those of its homomorphic image.The correspondence preserves prime, semiprime, and primary fuzzy ideals/kideals/h-ideals. Fuzzifications introduced in this paper can be labeled as "proper fuzzifications".

Preliminaries
2.1.Ideals of a Semiring.In the following discussion, (S, +, •) stands for a semiring.That is, (S, +) is a commutative monoid having identity element 0 and (S, •) is a semigroup satisfying the following identities: a(b + c) = ab + ac, (a + b)c = ac + bc, and 0 • x = 0 = x • 0. A commutative semiring with unity is a semiring (S, +, •) such that (S, •) is a commutative monoid.We denote the identity element of (S, •) by 1.With abuse of notation, we denote (S, +, •) by S. A left ideal A of S is a nonempty set A which is closed under the addition of S and is such that, for all x ∈ S and a ∈ A we have is similarly defined.Whenever a statement is made about left ideals, it is to be understood that the analogous statement is made about right ideals.An ideal is one, which is both right and left ideal.A left ideal P is called prime left ideal, if it satisfies the following conditions: (ii) for all left ideals A & B of S, we have It is natural to call P a k-prime (resp., h-prime) left ideal, if the condition (I) holds for left k-ideals (resp., h-ideals) A and B.
Clearly, every prime left ideal is k-prime and every kprime left ideal is h-prime.However, as will be seen in Example 1, the reverse implications, in general, are not true.
Example 1.(a) If S = {0, α, β, 1} is the Boolean lattice of four elements, then 0 is not a k-prime ideal, as the condition (I) fails for k-ideals A = {0, α} and B = {0, β}.However, S being the only h-ideal of S, 0 is h-prime.Clearly, 0 is neither prime nor an h-ideal.
(b) Consider the semiring S = {0, 1, 2, 3}, where the binary operations ⊕ and ⊗ are defined as follows: a ⊕ b = Min{a + b, 3} and a ⊗ b = Min{ab, 3}.One can easily see that S has only three proper ideals, namely, 0, A = {0, 2, 3}, and B = {0, 3}.Since we have AA ⊆ B and A / ⊆B, B is not a prime ideal.However, 0 and S being the only k-ideals of S, one can see that B is a k-prime ideal.Again, B is neither prime nor a k-ideal.
Proposition 2 (see [5,7]).If S is a semiring and A and B are left ideals of S, then k(AB) =

k(k(A)k(B)) and h(AB) = h(h(A)h(B)), where k(A) and h(A), respectively, denote kclosure and h-closure of A.
Using Proposition 2 we get the following.Proof.We prove the statement: "(b) implies (c)", for h-ideals.
Suppose P is a proper h-prime left h-ideal such that aSb ⊆ P for a, b ∈ S. Clearly, we have SaSb ⊆ P. Our first claim is that for A = h(Sa) and B = h(Sb), where h(I) stands for the h-closure of a left ideal I of S, we have AB ⊆ P. Suppose we have x ∈ A and y ∈ B. Then for some s, t, u, v, z, z , in S, we have x + sa + z = ta + z and y + ub + z = vb + z and, therefore, the equalities: ay + aub + az = avb + az , and xy + say + zy = tay + zy.As aub and avb are elements of P and P is an h-ideal, ay is in P. Therefore, say, tay and consequently, xy are in P. It, then, follows that AB ⊆ P and, P being h-prime, we have either A ⊆ P or B ⊆ P. Suppose A ⊆ P. If a is the left ideal generated by a, then we have a a ⊆ Sa and hence, h( a )h( a Using Zorn's Lemma one can prove the following.

Theorem 4. Every proper ideal (resp., k-ideal, h-ideal) of a commutative hemiring S with unity is contained in a prime ideal (resp., k-ideal, h-ideal) of S.
Theorem 5.If T is a multiplicatively closed set in a commutative hemiring S with unity, disjoint from an ideal (resp., k-ideal, h-ideal) I of S, then there exists a prime ideal (resp., k-ideal, hideal) P of S such that I ⊆ P and P ∩ T = ∅.

Prime Ideals of N.
In the hemiring N of nonnegative integers, obviously, an ideal I is a k-ideal if and only if it is an h-ideal.Moreover, I is a k-ideal if and only if I = nN for some n ∈ N. The prime k-ideals of N are either pN where p is a prime number in N or the zero ideal.For each prime p the ideal pN is a maximal k-ideal [19].Clearly, pN is not a maximal ideal of N. Proposition 6.Let p be a prime integer in N and P = N ∼ {1}.There is no prime ideal I of N such that pN ⊂ I ⊂ P.
Proof.We first prove that the proposition holds for p = 2. Assuming the contrary, let I be a prime ideal such that 2N ⊂ I ⊂ P. Let x be the smallest element of I ∼ 2N.Then x = 2n+ 1 for some positive integer n.
Since 2, 2n and 2n + 1 are in I and I is closed under addition, we have J ⊂ I. Clearly, if n = 1, then J = P. Therefore, we have n / = 1 and x ≥ 5. Consider y = 3.For sufficiently large value of s, we have y s is in J and hence, in I. Since I is a prime ideal, we have y ∈ I.This contradicts the assumption that x is the smallest element of I ∼ 2N.Therefore, I is not a prime ideal.
Consider a prime integer p ≥ 3 and a prime ideal I such that pN ⊂ I ⊂ P. Let x be the smallest element of I ∼ pN.Then, x = pn + r for some n ∈ N and r = 1, 2, . . ., p − 1.Consider x = pn + 1.Clearly, n / = 0 and thus, we have Observe that for all r = 1, 2, . . ., p − 1 we have (p − 1)pn + r = (p − (r + 1))pn + r(pn + 1).Therefore (p − 1)pn + r ∈ I.However, I contains pN and therefore, J. Now set y = 2. Since we assume I to be a prime ideal, we get y ∈ I ∼ pN.This contradicts the choice of x as the smallest element in I ∼ pN.Therefore, x / = pn + 1.Consider x = pn + r for 1 < r ≤ p − 1.Then, clearly, we have x ≥ 2. We claim that x / = 2.If x = 2, then 2N ⊆ I. Obviously, we have 2N / ⊂I.On the other hand, if 2N = I, then, we get the absurd result that pN ⊆ 2N for p / = 2. Now set, as before, y = 2 to get the contradiction to the assumption that x is the smallest element of I ∼ pN and complete the proof.Theorem 7. P = N ∼ {1} is the only prime ideal of N which is not a k-ideal (resp., an h-ideal).
Proof.One easily observes that P is a prime ideal and is not a k-deal.Let I be any other ideal of N, which is not a kideal.Clearly, then, we have 0 ⊂ I ⊂ P. Therefore, there exist x ∈ I such that 0 / = x / = 1.Let x = p α1 1 , . . ., p αn n be the prime factorization of x.If I is prime, there is at least one prime integer p in I. Therefore, we have pN ⊆ I ⊂ P. As I is not a kideal we have pN / = I.On the other hand, by Proposition 6 we cannot have a prime ideal I such that pN ⊂ I ⊂ P. Therefore, I is not a prime ideal.

Fuzzy Ideals of a
Semiring.Throughout this paper L stands for a complete Heyting algebra, that is, a complete lattice such that for all subsets T of L and all b An L-fuzzy left ideal J of S is an L-fuzzy set J : S → L such that for all a, b ∈ S the following conditions are satisfied: (i) J(a + b) ≥ J(a) ∧ J(b), (ii)J(ab) ≥ J(b).An L-fuzzy left ideal J of S is called an L-fuzzy left k-ideal, if the following condition is satisfied: a, b, z ∈ S.An L-fuzzy right ideal (resp., k-ideal, h-ideal) is similarly defined.Whenever a statement is made about L-fuzzy left ideals, it is to be understood that the analogous statement is made about an L-fuzzy right ideals.An L-fuzzy ideal is one, which is both L-fuzzy right and Lfuzzy left ideal.

Prime Fuzzy Ideals
We defined L-fuzzy prime h-ideal in [6].We extend the definition to L-fuzzy ideals and k-ideals.Definition 8.An L-fuzzy left ideal (resp., k-ideal, h-ideal) P of S is called a prime L-fuzzy left ideal (resp., k-ideal, h-ideal), if it is nonconstant and, for all a, b ∈ S and α ∈ L, the following condition is satisfied: Proposition 9 is proved for L-fuzzy left h-ideal in [6].
Let, further, P(a) ≥ α.Then, P(a Thus, P(S) is totally ordered.Conversely, let P(S) be totally ordered and ∧{P(asb) | s ∈ S} = P(a) ∨ P(b).Then, ( Therefore, P is a prime L-fuzzy ideal.This leads to the following elegant characterizations of prime fuzzy ideals.

Proposition 11. Let P be a nonconstant L-fuzzy ideal (resp., k-ideal, h-ideal) of S, and a, b ∈ S.
(1) P is prime if and only if ∧{P(asb) | s ∈ S} = P(a) ∨ P(b) and P(S) is totally ordered.
( The following example shows that the condition that P(S) is totally ordered is necessary for P to be prime.
Example 12. Let L = {0, α, β, 1} be the Boolean algebra of four elements.Consider the L-fuzzy ideal J : N → L defined as follows: Clearly, the L-fuzzy h-ideal J is not prime, though P(ab) = P(a) ∨ P(b) holds for all a, b ∈ N.

Advances in Fuzzy Systems
Remark 13.While fuzzifying the condition (I) of "primeness" stated in §2.1 three types of products of fuzzy left ideals A and B of S, are used in the literature: namely, AoB, Ao k B, and Ao h B [1-3, 5, 7].They are defined as follows: ( This was needed, because the problem of fuzzification of left ideals, left k-ideals, and left h-ideals were treated as three separate problems.Theorem 3 allows us to rope all the three in one and leads us to a compact characterization of primeness given in Proposition 11.
A semiprime fuzzy ideal, now defines itself.
Definition 14.An L-fuzzy left h-ideal J of S is called semiprime, if J is nonconstant and, for all a ∈ S and α ∈ L, the following condition is satisfied: It follows that a nonconstant L-fuzzy ideal (resp., k-ideal, hideal) I of S is semiprime if and only if ∧{I(asa) | s ∈ S} = I(a) for all s ∈ S. In case S is commutative hemiring with unity, the above equation is further simplified to I(a 2 ) = I(a).Analogues of Proposition 9 and Corollary 10 can easily be proved.

Theorem 15. Every nonconstant fuzzy ideal (resp., k-ideal, h-ideal) of a commutative ring with unity is contained in a minimal prime fuzzy ideal (resp., k-ideal, h-ideal).
Proof.As usual we prove the result for fuzzy h-ideals.Let J be a nonconstant fuzzy h-ideal of a commutative ring S with unity and J = {x ∈ S | J(x) > J(1)}.Let P be a prime h-ideal containing J. Define a fuzzy ideal P : S → [0, 1] by Clearly, P is a prime fuzzy h-ideal containing J and, thus, the class C of all prime fuzzy h-ideals containing J is nonempty.We partially order C by reverse containment, that is, we define P ≤ P if and only if P ⊆ P for all P, P ∈ C, and consider a totally ordered subset {P λ | λ ∈ Λ} of C.Then, the set {P λ α | λ ∈ Λ} of the α-level cuts of P λ is a totally ordered set consisting of prime h-ideals (and possibly of S) for each

Prime Radicals of a Fuzzy Ideal
In this section, we assume S to be a commutative hemiring with unity.
Definition 17.If J is an L-fuzzy ideal of S, then the intersection of all prime L-fuzzy ideals (resp., k-ideals, hideals) of S containing J is called the prime (resp., k-prime, h-prime) radical of J.We denote it by r(J) (resp., r k (J), r h (J)).
If the set of prime L-fuzzy ideals (resp., k-ideals, h-ideals) of S containing J is empty, we define r(J) (resp., r k (J), r h (J)) to be χ S .Note that r(J), (resp., r k (J), r h (J)) is a semiprime fuzzy ideal (resp., k-ideal, h-ideal) containing J. Clearly r(J) ⊆ r h (J) ⊆ r k (J).However, the following examples show that strict containment holds.
Example 18.Let p be a prime integer.Consider α, β ∈ [0, 1] and β < α.Define a fuzzy set P : N → [0, 1] by By Proposition 9, P is a prime fuzzy ideal (also k-ideal and h-ideal), for all 0 ≤ β < α ≤ 1.We will call the fuzzy ideal P a prime fuzzy k-ideal induced by the prime number p and denote it by (pN) αβ .
We will call the fuzzy ideal Q a prime fuzzy ideal induced by the prime integer p and denote it by (pN) αβγ .Note that, in the light of Theorem 7, these are the only prime fuzzy ideals of N which are not fuzzy k-ideals.
Example 20.Consider a fuzzy ideal defined by J : N → [0, 1]: Advances in Fuzzy Systems 5 Let 0 ≤ α < 1 and O α be the fuzzy k-ideal defined by O α : N → [0, 1]: Let Clearly, X is the set of all prime fuzzy k-ideals of N containing J and Y is the set of all those prime fuzzy ideals containing J, which are not fuzzy k-ideals.Since r h (J) = r k (J) = ∩{P | P ∈ X} and r(J it is mundane to verify that r h (J) = r k (J) = O 0.5 and r(J) is the fuzzy ideal defined by r(J) : N → [0, 1]: Clearly, r(J) ⊂ r k (J) = r h (J).

Nil Radicals of a Fuzzy Ideal
In this section, we assume S to be a commutative hemiring with unity.
Recall that if I is an ideal of S, then its radical (also called nil radical) is defined as We define the fuzzy analogue of nil radical as follows.
Definition 22.If J is an L-fuzzy ideal of S, then the L-fuzzy set Through series of propositions we prove that, when L is totally ordered and J is a fuzzy k-ideal (resp., h-ideal) of S, so is Advances in Fuzzy Systems Clearly, if x ∈ P, then we have J(x) ≤ P(x).On the other hand, if x / ∈ P, then we have x / ∈ J α+ and J(x) ≤ α = P(x).Thus, we get J ⊆ P and consequently, r h (J) ⊆ P.
Hence, we have √ J = r h (J).
Corollary 29.Let L be a totally ordered set.If J is an L-fuzzy k-ideal (resp., h-ideal) of S, then, so is √ J.

Correspondence Theorems
In this section, f : S → S is a homomorphism of hemirings, J is an L-fuzzy left ideal of S, and J is an L-fuzzy left ideal of S .
In [20,Proposition 3.11], Zhan claims that if J is an Lfuzzy h-ideal with sup property, then f (J) is an L-fuzzy hideal of f (S).The following example does not substantiate the claim.
Example 30.Let S be the hemiring given in Example 1 (b), N be the hemiring of non-negative integers, and f : N → S be the epimorphism given by f (x) = min{x, 3} for all x ∈ N.
Define a mapping J : N → [0, 1] by Since f (x) = x for all x ≤ 3 and f (x) = 3 for all x ≥ 3, it can be verified that f (J)(0) = 1, f (J)(2) = f (J)(3) = 1/2, and f (J)(1) = 0.One can readily see that J is an L-fuzzy h-ideal with sup property; but f (J) is not an L-fuzzy h-ideal.For, we have 1 . Example 30 raises a natural question: What are the sufficient conditions for a homomorphic image of an h-ideal (resp., k-ideal) to be an h-ideal (resp., k-ideal)?In order to answer this question, we introduce the following definition.
We leave it to the reader to prove that an f -compatible fuzzy left ideal is f -invariant.
Proposition 32.Let f : S → S be a homomorphism of hemirings and J and J L-fuzzy left ideals of S and S , respectively.Then, the following statements hold.
(1) f −1 (J ) is an f-invariant L-fuzzy left ideal of S.
(2) If J is an L-fuzzy left k-ideal, then so is f −1 (J ). ( Proof.We prove (4) and ( 5).If J is f -invariant and x ∈ S, then it is obvious that f This leads to the following correspondence theorem for L-fuzzy left k-ideals and h-ideals.
Theorem 33.Let f : S → S be an epimorphism of hemirings.
(1) There is one-to-one correspondence between the set of L-fuzzy left ideals (resp., k-ideals) of S and that of finvariant L-fuzzy left ideals (resp., k-ideals) of S.
(2) There is one-to-one correspondence between the set of L-fuzzy left h-ideals of S and that of f-compatible Lfuzzy left h-ideals of S.
Proof.Suppose Jand J are L-fuzzy left ideals of S and S .By Proposition 32, the correspondence is given by J ↔ f (J) and J ↔ f −1 (J ).We only need to verify that, when J is an Lfuzzy left ideal (resp., k-ideals, h-ideal), then so is f (J), under the conditions specified for J.A reader may easily prove that, when J is f-invariant, f (J) is an L-fuzzy left ideal.Let, moreover, J be an L-fuzzy k-ideal, x + a = b , f (x) = x , f (a) = a , and f (b) = b .Then f (x + a) = f (b) and therefore, we have J(x + a) = J(b).Let α = J(a) ∧ J(b) and consider J α .Clearly, a, b ∈ J α .Since J(x + a) = J(b), we have x + a ∈ J α and J α being a k-ideal x ∈ J α .Therefore, J(x) ≥ J(a) ∧ J(b).But, by Proposition 32 (4), this inequality is equivalent to f (J)(x ) ≥ f (J)(a ) ∧ f (J)(b ).Thus, f (J) is an L-fuzzy left k-ideal.
On similar lines, one can prove that, when J is an fcompatible L-fuzzy left h-ideal of S, f (J) is an L-fuzzy left h-ideal of S .

Primary Fuzzy Ideals
In this section, we assume that S is a commutative hemiring with unity.
Recall that an ideal Q of a hemiring S is primary, if (i) Q / = S and (ii) xy ∈ Q ⇒ x ∈ Q or y n ∈ Q for some positive integer n.
We define primary fuzzy ideal as follows.
Definition 34.A nonconstant L-fuzzy ideal/k-ideal/h-ideal Q of S is primary, if Q(xy) = Q(x) or Q(xy) ≤ Q(y n ) for some positive integer n.
The following propositions are immediate consequences of Definition 34.

Theorem 3 .
Let P be a proper left k-ideal (resp., h-ideal) of a semiring S. The following statements are equivalent.(a) P is prime.(b) P is k-prime (resp., h-prime).(c) For all a, b ∈ S, aSb ⊆ P implies a ∈ P or b ∈ P.
Corollary 10.A left ideal (resp., k-ideal, h-ideal) P of S is prime if and only if its characteristic function χ P is an Lfuzzy prime left ideal (resp., k-ideal, h-ideal) for every completeHeyting algebra L.
) Proposition 9. A nonconstant L-fuzzy left ideal (resp., k-ideal, h-ideal) P of S is prime if and only if its every nonempty level cut of P is either a prime left ideal (resp., k-ideal, h-ideal) of S or S itself.
is an upper bound of the family {P λ | λ ∈ L}, C has a maximal element which, clearly, is a minimal prime fuzzy h-ideal containing J.