2-Prime ideals and their applications

Abstract

This paper introduces the notion of [Formula: see text]-prime ideals, and uses it to present certain characterization of valuation rings. Precisely, we will prove that an integral domain [Formula: see text] is a valuation ring if and only if every ideal of [Formula: see text] is [Formula: see text]-prime. On the other hand, we will prove that the normalization [Formula: see text] of [Formula: see text] is a valuation ring if and only if the intersection of integrally closed 2-prime ideals of [Formula: see text] is a 2-prime ideal. At the end of this paper, we will give a generalization of some results of Gilmer and Heinzer by studying the properties of domains in which every primary ideal is an integrally closed 2-prime ideal.

$(\delta,2)$-primary ideals of a commutative ring

“…(⇐): If a proper ideal I of a ring R is a 2-prime, then √ I is prime in [6], Proposition 1.3, statement (1). Thus, we have √ I = I for all ideals I of R. Therefore R is von Neumann regular.…”

“…For any undefined notation or terminology, see [3], [7] or [10]. In [6], the authors introduced 2-prime ideals and gave the basic properties and some applications of the concept on valuation rings. A proper ideal I of R is called 2-prime if whenever a, b ∈ R and ab ∈ I then either a 2 ∈ I or b 2 ∈ I.…”

N Sequence Prime Ideals

“…] is neither -sequence prime nor quasi prime. Definition 3.5 [4]. A proper ideal of is -prime (resp.…”

2-Prime Hyperideals

“…Dasgupta extended the prime and primary hyperideals in multiplicative hyperrings in [16]. Beddani and Messirdi [10] introduced a generalization of prime ideals called 2-prime ideals and this idea is further generalized by Ulucak and et. al.…”