On Nagata rings with amalgamation properties

Let [Formula: see text] and [Formula: see text] be two commutative rings with unity, let [Formula: see text] be an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. In this paper, we give a characterization for the amalgamated algebra [Formula: see text] to be a Nagata ring, a strong S-domain, and a catenarian. Also, we investigate the conditions that the ring of Hurwitz series over [Formula: see text] has a complete comaximal factorization.

A Note on Weakly S-Noetherian Rings

Let R be a commutative ring with identity and S a (not necessarily saturated) multiplicative subset of R. We call the ring R to be a weakly S-Noetherian ring if every S-finite proper ideal of R is an S-Noetherian R-module. In this article, we study some properties of weakly S-Noetherian rings. In particular, we give some conditions for the Nagata’s idealization and the amalgamated algebra to be weakly S-Noetherian rings.

Comparison of local cohomlogy modules of a ring and its amalgamated algebra along a given ideal

Let [Formula: see text] be a commutative Noetherian ring and let [Formula: see text] and [Formula: see text] be ideals of [Formula: see text]. The main purpose of this paper is to compare of the finiteness properties of [Formula: see text] and [Formula: see text], where [Formula: see text] is the [Formula: see text]th local cohomology module functor with respect to [Formula: see text] and [Formula: see text] is the amalgamated of [Formula: see text] along [Formula: see text].