Abstract The purposes of this paper are to introduce generalizations quasi-prime ideals the context $$\phi $$ ? -quasi-prime ideals. Let : {\mathcal {I}}(S) \rightarrow \cup \left\{ \emptyset \right\} : I ( S ) </m...

Let R be a local ring of bounded module type. It is shown that R is an almost maximal valuation ring if there exists a non-maximal prime ideal J such that R/J is an almost maximal valuation domain. We deduce from this that R is almost maximal if one of the following conditions is satisfied: R is a Q-algebra of Krull dimension ≤ 1 or the maximal ideal of R is the union of all non-maximal prime i...

The notions of a S fuzzy ∧ sub semi lattice, a S fuzzy ideal and a S fuzzy prime ideal of a bounded lattice with truth values in a bounded ∧ sub semi lattice S are introduced which generalize the existing notions with truth values in a unit interval of real numbers. Finally, S fuzzy prime ideal theorem is proved. 2010 AMS Classification: 03G10, 46H10, 06D50, 08A72

Abstract We show that every sufficiently large integer is a sum of prime and two almost squares, also smooth number squares. The such representations the expected order magnitude. likewise treat shifted primes $$p-1$$ p - 1 as sums methods involve combi...