In this short note, we prove that almost all integers N satisfying N ≡ 3 (mod 24) and 5 -N or N ≡ 4 (mod 24) is the sum of three or four almost equal prime squares, respectively: N = p21 + · · ·+ p 2 j with |pi − (N/j) 1/2| ≤ N1/2−9/80+ε for j = 3 or 4 and 1 ≤ i ≤ j.

Let 9t and © be commutative rings such that © contains, and has the same identity element as, 9Î. If p and $ are prime ideals in SK and © respectively such that ^P\9t = p then we shall say that $ lies over, or contracts to, p. If over every prime ideal in dt there lies a prime ideal in ©, we shall say that the "lying-over" theorem holds for the pair of rings 9Î and ©. Suppose now that q and p a...

Double Boolean algebras are algebras (D,u,t, , ,⊥,>) of type (2, 2, 1, 1, 0, 0). They have been introduced to capture the equational theory of the algebra of protoconcepts. A filter (resp. an ideal) of a double Boolean algebra D is an upper set F (resp. down set I) closed under u (resp. t). A filter F is called primary if F 6= ∅ and for all x ∈ D we have x ∈ F or x ∈ F . In this note we prove t...

In this paper, we first introduce the concepts of weakly prime and weakly quasi-prime fuzzy left ideals of an ordered semigroup S. Furthermore, we give some characterizations of weakly prime and weakly quasi-prime fuzzy left ideals of an ordered semigroup S by the ordered fuzzy points and fuzzy subsets of S. Keywords—Ordered semigroup; ordered fuzzy point; weakly prime fuzzy left ideal; weakly ...

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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...