Semiring is one of the universal algebras which is a generalization not only of ring but also of distributive lattice. The notion of semiring was introduced by an author in

The purpose of this section is to provide a concise but reasonably complete exposition of the background material for the subsequent sections of this paper.

Definition 2.1.

Definition 2.2.

(i)

(ii)

(iii)

(iv)

Definition 2.5.

Definition 2.6.

Definition 2.7.

Definition 2.8.

Definition 2.9.

Definition 2.10. Let

Definition 2.11. Let

Definition 2.12. Let

A fuzzy ideal of a

Definition 2.14. Let

Remark 2.15. Throughout this paper,

In this section, we investigate some fundamental results of fuzzy ideals of semirings

We start this section with definitions required to construct a

Definition 3.1. A fuzzy ideal

OR

A fuzzy ideal

Example 3.2. Let

and

Then

Definition 3.3. Let

Proof. Let

Therefore,

The following theorems are proved in

(i)

(ii)

(iii)

(i) Any fuzzy subset

(ii) If

(iii)

(i)

(ii)

(iii)

(iv) If

(v)

Proof. The proof of (i), (ii) and (v) are simple and straightforward.

(iii) Since

Thus,

(iv) Using (i), we have

Let R be a Γ− semiring. Then FLI(R), FRI(R) and FI(R) denote respectively the set of all fuzzy left ideals, set of all fuzzy right ideals and set of all fuzzy ideals of R.

Now the following results are proved in

Let

Let I be an ideal of

(i) If

(a)

(b)

(ii) For any ideal

(iii) R is centreless if and only if

(ii) This follows from definition 3.3, since for

Let

Proof. The result follows directly by theorem 3.14((i)(b),(iii)) and theorem 3.13.

Now, fix

Finally, in this section we have

Proof. The (i) and (ii) part follows from Theorem

In this section, we observe that a fuzzy ideal of a

Definition 4.1. Let

We now state the following lemma and theorem, proof of which are analogous to the corresponding theorems in semirings

As an application of the above result, we have

Proof. Let

Using theorem 3.7(ii), we find that

Now we show that

In this paper, we establish a special class of fuzzy ideals of a

This work has been presented in “International conference on Recent Strategies in Mathematics and Statistics (ICRSMS-2022), Organized by the Department of Mathematics of Stella Maris College and of IIT Madras during 19 to 21 May, 2022 at Chennai, India. The Organizer claims the peer review responsibility.

The authors are highly grateful to the referee for his/her careful reading, valuable suggestions and comments, which helped us to improve the presentation of this paper.