Let R be a commutative integral domain with quotient field K and let P be a nonzero strongly PRIME IDEAL of R. We give several characterizations of such IDEALs. It is shown that (P : P) is a valuation domain with the unique maximal IDEAL P. We also study when P-1 is a ring. In fact, it is proved that P-1 = (P : P) if and only if P is not invertible. Furthermore, if P is invertible, then R = (P : P) and P is a principal IDEAL of R.

In this paper, PRIME IDEAL bundles and semi-PRIME and irreducible IDEAL bundles of a Lie algebra bundle are defined and their relation with PRIME IDEAL bundles is studied.

The aim of this paper is to establish the PRIME state IDEAL theorem in state residuated lattices (SRLs). We study the state IDEALs lattice SI(L) of a state residuated lattice (L, φ, ) and prove that it is a complete Brouwerian lattice in which the meet and the join of any two compact elements are compact (coherent frame). We characterize the notion of PRIME state IDEALs in SRLs. In addition, we establish the condition for which the lattice SI(L) is a Boolean algebra.

Today, literature and poetry have a more comprehensive definition than mere literary expression with academic meanings and tools. Literature is the collection of all the scientific abilities of the language in the field of structure and richness. The artistic creativity of the poet and the employment of excellent compositions and themes lead to the production of a literary and artistic work, which can be found in the excitement and expansion of the audience. The aim of this research is to reach the communication network of all the factors related to the language and the appearance of the word, which not only carries the meaning, but also creates a different meaning and concept from the apparent meaning that results from the phonetic, lexical or visual mechanism. A network that is referred to as "artwork". Artistic device (PRIME) is a meta-array approach to the aesthetic tools of language in poetry, which formalists consider to be the art of alienating language from the mind of the audience.

In this paper we introduce the notion of multiplication IDEALs in Γ-rings and we obtain some characterizations for multiplication IDEALs in Γ-rings.

In this paper, we introduce the concepts of J-PRIME IDEALs and MJ-IDEALs in posets, and obtain some of their interesting characterizations in posets. Furthermore, we discuss the properties of J-IDEALs that are analogous to J-PRIME IDEALs and MJ-IDEALs in posets. Finally, we establish a set of equivalent conditions for an IDEAL in a poset P containing an IDEAL J is an J-IDEAL, and for a semi-PRIME IDEAL J to be an MJ-IDEAL of P.

In this paper, we de ne the notion of minimal PRIME IDEALs of hoops and investigate some properties of them. Then by using the notion of annihilators, we study the relation between minimal PRIME IDEALs and annihilators. Also, we introduce the notion of zero divisors elements of hoops and prove that the set of all zero divisors of hoops is a union of all minimal PRIME IDEALs of hoop. Finally, by using the notions of minimal PRIME IDEALs and maximal IDEALs of hoop, we introduce two new IDEALs as p-IDEAL and m-IDEAL. Then we study some properties of them and investigate the relation between them and prove that every p-IDEAL of semi-simple hoop is an m-IDEAL of it.

Let G be a , nitely generated abelian group and M be a G-graded A-module. In general, G-associated PRIME IDEALs to M may not exist. In this paper, we introduce the concept of G-attached PRIME IDEALs to M as a generalization of G-associated PRIME IDEALs which gives a connection between certain G-PRIME IDEALs and G-graded modules over a (not necessarily G-graded Noetherian) ring. We prove that the G-attached PRIME IDEALs exist for every nonzero G-graded module and this generalization is proper. We transfer many results of G-associated PRIME IDEALs to G-attached PRIME IDEALs and give some applications of it.