Narrow Results

The study of prime ideals has been an area of active research. In recent past a considerable work has been done in this direction. Associated prime ideals and minimal prime ideals of certain types of Ore extensions have been characterized.

In this paper a relation between completely prime ideals of a ring R and those of R[x; σ, δ] has been given; σ is an automorphisms of R and δ is a σ-derivation of R. It has been proved that if P is a completely prime ideal of R such that σ(P) = P and δ(P) ⊆ P, then P[x; σ, δ] is a completely prime ideal of R[x; σ, δ]. It has also been proved that this type of relation does not hold for strongly prime ideals.

In this paper, we describe some different variations of prime ideals in the context of rings of continuous functions such as strongly prime ideals, almost prime ideals, $n$-absorbing ideals, and 2-prime ideals. We characterize the strongly prime ideals of $C(X)$. We prove that an ideal $I$ of $C(X)$ is almost prime if and only if it is semiprime or equivalently if and only if it is an $(m,n)$-closed ideal, where $m$ and $n$ are positive integers. In [On $n$-absorbing ideals of commutative rings, Comm. Algebra 39 (2011) 1646–1672], Anderson and Badawi conjectured that every $n$-absorbing ideal of a commutative ring is strongly $n$-absorbing. We determine the $n$-absorbing ideals of $C(X)$ and show that the conjecture holds in the case of rings of continuous functions. We also characterize the rings of continuous functions $C(X)$ in which every pseudoprime ideal is 2-prime.

Kaplansky introduced the concept of the K-rings, concerning the commutativity of rings. In this paper, we concentrate on a property of K-rings, introducing the concept of the strongly NI rings, which is stronger than NI-ness. We first examine the relations among the concepts concerned with K-rings and strongly NI rings, constructing necessary examples in the process. We also show that strong NI-ness is a hereditary radical property.

In 2013, Bergamaschi and Santiago proposed Strongly Prime Fuzzy(SP) ideals for commutative and noncommutative rings with unity, and investigated their properties. This paper goes a step further since it provides the concept of Strongly Prime Radical of a fuzzy ideal and its properties are investigated. It is shown that Zadeh's extension preserves strongly prime radicals. Also, a version of Theorem of Correspondence for strongly prime fuzzy ideals is proved.