Noetherian spectrum condition and the ring 𝒜[X

Abstract

Let [Formula: see text] be an ascending chain of commutative rings with identity and let [Formula: see text] be the ring of polynomials with coefficient of degree [Formula: see text] in [Formula: see text] for each [Formula: see text] In the first part of this paper, we give a necessary and sufficient conditions for the ring [Formula: see text] to satisfy Noetherian spectrum (a ring [Formula: see text] is said to have Noetherian spectrum if [Formula: see text] satisfies the Ascending Chain Condition (ACC) on radical ideals). In particular, we extend classical results proved by Ohm and Pendleton in [J. Ohm and R. Pendleton, Rings with Noetherian spectrum, Duke Math. J. 35 (1968) 631–639] and by Hizem in [S. Hizem, Chain conditions in rings of the form A + XB[X] and A + XI[X], in Commutative Algebra and its Applications (de Gruyter, 2009), pp. 259–274]. The second part of this paper is motivated by the interesting results proved on uniformly [Formula: see text]-Noetherian rings by Kim et al. in [W. Qi, H. Kim, F. Wang, M. Chen and W. Zhao, Uniformly S-Noetherian rings (submitted)]. We introduce the concept of uniformly [Formula: see text]-Noetherian spectrum and study its properties. Let [Formula: see text] be a commutative ring and [Formula: see text] a multiplicative subset of [Formula: see text] We say that [Formula: see text] has uniformly[Formula: see text]-Noetherian spectrum if that there exists an [Formula: see text] such that for any ideal [Formula: see text] of [Formula: see text], [Formula: see text] for some finitely generated subideal [Formula: see text] of [Formula: see text] We give the Eakin–Nagata–Formanek theorem for the uniformly [Formula: see text]-Noetherian spectrum condition. We also study the uniformly [Formula: see text]-Noetherian uniformly properties on several ring constructions (Nagata’s idealization and amalgamated algebras).