Variations of primeness of ideals in rings of continuous functions

Abstract

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, [Formula: see text]-absorbing ideals, and 2-prime ideals. We characterize the strongly prime ideals of [Formula: see text]. We prove that an ideal [Formula: see text] of [Formula: see text] is almost prime if and only if it is semiprime or equivalently if and only if it is an [Formula: see text]-closed ideal, where [Formula: see text] and [Formula: see text] are positive integers. In [On [Formula: see text]-absorbing ideals of commutative rings, Comm. Algebra 39 (2011) 1646–1672], Anderson and Badawi conjectured that every [Formula: see text]-absorbing ideal of a commutative ring is strongly [Formula: see text]-absorbing. We determine the [Formula: see text]-absorbing ideals of [Formula: see text] and show that the conjecture holds in the case of rings of continuous functions. We also characterize the rings of continuous functions [Formula: see text] in which every pseudoprime ideal is 2-prime.