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Maji et al. introduced the concept of intuitionistic fuzzy soft sets, which is an extension of soft sets and intuitionistic fuzzy sets. In this paper, we apply the concept of intuitionistic fuzzy soft sets to rings. The concept of intuitionistic fuzzy soft rings is introduced and some basic properties of intuitionistic fuzzy soft rings are given. Intersection, union, AND, and OR operations of intuitionistic fuzzy soft rings are defined. Then, the deffinitions of intuitionistic fuzzy soft ideals are proposed and some related results are considered.

Uncertain or imprecise data are inherent and pervasive in many important applications in the areas such as economics, engineering, environment, social science, medical science, and business management. Uncertain data in those applications could be caused by data randomness, information incompleteness, limitations of measuring instrument, delayed data updates, and so forth. Due to the importance of those applications and the rapidly increasing amount of uncertain data collected and accumulated, research on effective and efficient techniques that are dedicated to modeling uncertain data and tackling uncertainties has attracted much interest in recent years and yet remained challenging at large. There have been a great amount of research and applications in the literature concerning some special tools like probability theory, (intuitionistic) fuzzy set theory, rough set theory, vague set theory, and interval mathematics. However, all of these have theirs advantages as well as inherent limitations in dealing with uncertainties. One major problem shared by those theories is their incompatibility with the parameterizations tools. Soft set theory [

At present, work on the extension of soft set theory is progressing rapidly. Maji et al. proposed the concept of fuzzy soft set [

From the above discussion, we can see that all of these works are based on Zadeh’s fuzzy sets theory which was generalized to intuitionistic fuzzy sets by Atanassov [

The purpose of this paper is to deal with the algebraic structure of ring by applying intuitionistic fuzzy soft theory. We introduce the notion of intuitionistic fuzzy soft ring and study some of its characterization of operations and algebraic properties.

In this section, for the sake of completeness, we first cite some useful definitions and results.

A fuzzy subset

Let

Let

An intuitionistic fuzzy set

Let

Let

An intuitionistic fuzzy soft set is a parameterized family of intuitionistic fuzzy subsets of

In general, for all

If for all

Let

Then, the family

Let

We denote the above inclusion relationship by

Let

Let

The intersection of two intuitionistic fuzzy soft sets

The union of two intuitionistic fuzzy soft sets

In contrast with the above definitions of union and intersection of intuitionistic fuzzy soft sets, we may sometimes adopt different definitions of union and intersection given as follows.

Let

Let

A fuzzy set

A pair

For two intuitionistic fuzzy soft rings

for each

Two intuitionistic fuzzy soft rings

Let

The proof is straightforward.

The union of two intuitionistic fuzzy soft rings

This is denoted by

If

For any

Thus, in any cases, we have

Therefore,

The intersection of two intuitionistic fuzzy soft rings

This is denoted by

If

The proof is straightforward.

Let

If

For all

The case for

Consider the ring

Let

Obviously,

Let

If

The proof is straightforward.

The following theorem is a generalization of previous results.

Let

If

Let

Suppose that

For all

The proof of other cases is similar.

Let

If

Let

Similarly, we have

In a similar way, we prove that

Let

One has

For any intuitionistic fuzzy soft ideals

Since

Since

We obtain a similar result for

For any fuzzy soft ideals

Assume that

Since

Let

Let

Let

If there exists an intuitionistic fuzzy soft ring homomorphism between

Now, we show that the homomorphic image and preimage of an intuitionistic fuzzy soft ring are also intuitionistic fuzzy soft rings.

Let

The image of

The preimage of

If

Let

Let

The proof is straightforward.

Let

The proof is straightforward.

In this work the theoretical point of view of intuitionistic fuzzy soft sets in ring and ideal is discussed. The work is focused on intuitionistic fuzzy soft rings and fuzzy soft ideals. These concepts are basic structures for improvement of fuzzy soft set theory. One can extend this work by studying other algebraic structures.

The authors are highly grateful to the referees for their valuable comments and suggestions for improving the paper.