Solution: The ring F [x] of polynomials with coefficients in a field F is a P.I.D. Each prime ideal is generated by a monic, irreducible polynomial. Assume there are only a finite number of prime ideals generated by the polynomials f1, . . . , fn and let f(x) = 1+f1(x) · · · fn(x). No fi divides f , hence f is also irreducible. This contradicts the assumption that all the prime ideals were gene...

We consider the intuitionistic fuzzification of the concept of prime ideals in commutative BCK-algebras, and investigate some of their properties. We show that if P is a prime ideal of commutative BCK-algebra X iff ∼ P = 〈XP , − XP 〉 is an intuitionistic fuzzy prime ideal of X. We also prove that An IFS A = 〈μA, λA〉 of commutative BCK-algebra X is an intuitionistic fuzzy prime ideal of X if and...

Pseudomonomials and ideals generated by pseudomonomials (pseudomonomial ideals) are a central object of study in the theory of neural rings and neural codes. In the setting of a polynomial ring, we define the polarization operation ρ sending pseudomonomials to squarefree monomials and a further polarization operation P sending pseudomonomial ideals to squarefree monomial ideals. We show for a p...

The main purpose of this paper is to continue the study of uniform strong primeness on fuzzy setting. A pure fuzzy notion of this structure allows us to develop specific fuzzy results on USP (uniformly strongly prime) ideals over commutative and noncommutative rings. Besides, the differences between crisp and fuzzy setting are investigated. For instance, in crisp setting an ideal I of a ring R ...