An elementary argument shows that a prime element must be irreducible, but the converse fails (see equation (∗) and its accompanying description in Section 4). Moreover, it is easy to verify that a prime element p has the nice property that any atom which divides a power pk of p must be an associate of p. We call an atom which has this property a strong atom (in the literature, these atoms have...

We explore the classical pattern avoidance question in the case of irreducible permutations, i.e., those in which there is no index i such that σ(i + 1) − σ(i) = 1. The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary nu...

In this paper, we study some properties of φ-prime submodules andwe give another charactrization for it. For given N and K a moduleM with ⊆ N, prove that is submodule if only N/Kis φ_K-prime submodule. Finally, show any finite sum isφ-prime.

As further generalizations of Uq(sl2), an interesting class of algebras Uq(f(K,H)) was introduced and studied in [10]. In this paper, we further study these algebras by realizing them as Hyperbolic algebras. Such a realization yields a natural construction of interesting families of irreducible representations of Uq(f(K,H)) by using methods in non-commutative algebraic geometry([16]). We also i...

Regarding non-unique factorization of integer-valued polynomials over a discrete valuation domain (R,M) with finite residue field, it is known that there exist absolutely irreducible elements, is, elements all whose powers factor uniquely, and non-absolutely elements. We completely constructively characterize the among split polynomials. They correspond bijectively to sets, which we call balanc...

The notion of intuitinitic (T, S)-fuzzy submodule in Module is introduced, and related properties are investigated. Mathematics Subject Classification: 08A72, 13C99

in this paper, the notion of fully idempotent modules is defined and it is shown that this notion inherits most of the essential properties of the usual notion of von neumann's regular rings. furthermore, we introduce the dual notion of fully idempotent modules (that is, fully coidempotent modules) and investigate some properties of this class of modules.

let $r$ be a commutative ring with identity and let $m$ be an $r$-module. a proper submodule $p$ of $m$ is called strongly prime submodule if $(p + rx : m)ysubseteq p$ for $x, yin m$, implies that $xin p$ or $yin p$. in this paper, we study more properties of strongly prime submodules. it is shown that a finitely generated $r$-module $m$ is artinian if and only if $m$ is noetherian and every st...