In this paper, we introduce and study graded weakly 1-absorbing prime ideals in commutative rings. Let \(G\) be a group \(R\) \(G\)-graded ring with nonzero identity \(1\neq0\). A proper ideal \(P\) of is called if for each nonunits \(x,y,z\in h(R)\) \(0\neq xyz\in P\), then either \(xy\in P\) or \(z\in P\). We give many properties characterizations ideals. Moreover, investigate under homomorph...

We introduce the notions of interval-valued fuzzy prime ideals, interval-valued fuzzy completely prime ideals and intervalvalued fuzzy weakly completely prime ideals. And we give a characterization of interval-valued fuzzy ideals and establish relationships between interval-valued fuzzy completely prime ideals and interval-valued fuzzy weakly completely prime ideals.

The concept of q , -fuzzy ternary subsemigroup (left, right, lateral, quasi-, bi-) ideal of a ternary semigroup is introduced and several related properties are investigated. We give some characterizations of regular and weakly regular ternary semigroups by q , -fuzzy left (right, lateral) ideals, q , -fuzzy quasi-ideals and q , -fuzzy bi-(generalized bi-) ideals.

This paper introduce and study weakly 1-absorbing prime ideals in commutative rings. Let $$A $$ be a ring with nonzero identity $$1\ne 0.$$ A proper ideal $$P of is said to if for every nonunits $$x,y,z\in $$0\ne xyz\in P, then $$xy\in P$$ or $$z\in P. In addition give many properties characterizations ideals, we also determine rings which prime. Furthermore, investigate C(X), the continuous...

A cyclotomic polynomial Φn(x) is said to be ternary if n = pqr with p, q and r distinct odd prime factors. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Eli Leher showed in 2007 that neighboring ternary cyclotomic coefficients differ by at most four. We show that, in fact, they differ by at most one. Consequently, ...