In this paper we introduce the concept of weakly prime ternary subsemimodules of a ternary semimodule over a ternary semiring and obtain some characterizations of weakly prime ternary subsemimodules. We prove that if $N$ is a weakly prime subtractive ternary subsemimodule of a ternary $R$-semimodule $M$, then either $N$ is a prime ternary subsemimodule or $(N : M)(N : M)N = 0$. If $N$ is a $Q$-...

‎in this paper‎, ‎the notion of the radical of a filter in‎ ‎residuated lattices is defined and several characterizations of‎ ‎the radical of a filter are given‎. ‎we show that if f is a‎ ‎positive implicative filter (or obstinate filter)‎, ‎then‎ ‎rad(f)=f‎. ‎we proved the extension theorem for radical of filters in residuated lattices‎. ‎also‎, ‎we study the radical‎ ‎of filters in linearly o...

Let $R$ be a commutative ring with identity‎. ‎A proper ideal $P$ of $R$ is a $(n-1,n)$-$Phi_m$-prime ($(n-1,n)$-weakly prime) ideal if $a_1,ldots,a_nin R$‎, ‎$a_1cdots a_nin Pbackslash P^m$ ($a_1cdots a_nin Pbackslash {0}$) implies $a_1cdots a_{i-1}a_{i+1}cdots a_nin P$‎, ‎for some $iin{1,ldots,n}$; ($m,ngeq 2$)‎. ‎In this paper several results concerning $(n-1,n)$-$Phi_m$-prime and $(n-1,n)$-...

Let G=Cn×Cn, where Cn denotes a cyclic group of order n, and let k∈[0,n−1]. We study the structure sequences terms from G with maximal length |S|=2n−2+k that fail to contain nontrivial zero-sum subsequence at most 2n−1−k. For k≤1, this is inverse question for Davenport Constant. k=n−1, η(G) invariant concerning short subsequences. The in both these cases (known respectively as Property B C) was...

in this paper we introduce the concept of weakly prime ternary subsemimodules of a ternary semimodule over a ternary semiring and obtain some characterizations of weakly prime ternary subsemimodules. we prove that if $n$ is a weakly prime subtractive ternary subsemimodule of a ternary $r$-semimodule $m$, then either $n$ is a prime ternary subsemimodule or $(n : m)(n : m)n = 0$. if $n$ is a $q$-...

According to Chajda and Eigenthaler ([1]), a d-lattice is a bounded lattice L satisfying for all a, c ∈ L the implications (i) (a, 1) ∈ θ(0, c) → a ∨ c = 1; (ii) (a, 0) ∈ θ(1, c) → a ∧ c = 0; where θ(x, y) denotes the least congruence on L containing the pair (x, y). Every bounded distributive lattice is a d-lattice. The 5-element nonmodular lattice N 5 is a d-lattice. Theorem 1 A bounded latti...

In this paper, we introduce and study the stable topology on the set of prime filters of a bounded 0-distributive lattice. The stable topology is a subtopology of the hull kernel topology on the set of prime filters of a bounded 0-distributive lattice. Sufficient condition is given under which the hull kernel topology and stable topology coincide on the set of prime filters (the set of maximal ...

Let $G$ be a monoid with identity $e$. In this paper, first we introduce the notions of $G$-graded hyperrings, graded hyperideals and graded hyperfields in the sense of Krasner hyperring $R$. Also, we define the notion of a greded $R$-hypermodules and some examples are presented. Then we investigate graded maximal, graded prime and graded primary hyperideals of a graded hyperring $R$. Finally, ...