We discuss a fuzzy result by displaying an example that shows how a classical argument fails to work when one passes from classical logic to fuzzy logic. Precisely, we present an example to show that, in the fuzzy context, the fact that the supremum is naturally used in lieu of the union can alter an argument that may work in the classical context.

Rosenfeld in 1971 was the first classical algebraist to introduce fuzzy algebra by writing a paper on fuzzy groups [

In the field of commutative ring, it is customary to use star operations not only to generalize classical domains, but also to produce a common treatment and deeper understanding of those domains. Some of the instances are the notion of Prüfer

In this note, we focus on some classical arguments of multiplicative ideal theory that do not hold in the fuzzy context. The example chosen is to infer that what appears to be pretty simple and even rather easy in the context of classical logic may not be true in the fuzzy context. So, one challenge of fuzzification is to detect any defect or incongruous statement that may first appear benign but is a real poison in the argument used to prove fuzzy statements. Precisely, in our example, we display the difficulty in how the natural definition in the fuzzy context may make it a little bit more challenging to work with in comparison with its equivalent classical definition. For an overview of all definitions of fuzzy submodules, fuzzy ideals, and fuzzy (semi)star operations (of finite character), the reader may refer to [

Recall that an integral domain

A

A

Recall from [

Let

If

(1) It is clear by definition that

(2) The constant map

(3) Let

for any

Recall from [

Let

Let

We now produce an example to prove that the fuzzy counterpart statement is false. Note that the reason why the counterpart may be false is clearly the fact that the union in the fuzzy context is the supremum. So, the real challenge here is to construct a counterexample that will clearly justify the wrongness of the argument.

Let

To demonstrate that

Now, let us consider

Now, let us demonstrate the false argument. Let us consider

The proof of the classical argument holds due to the fact that the classical union is involved allowing the choice of a finitely generated

We must also note that the fuzzy counterpart statement is the natural one that grasps some thoughts about the context in which the crisp result can be extended. In fact, the condition of union preserving of fuzzy star operation, that is,

The author declares that there are no competing interests regarding the publication of this paper.