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.

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

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.