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.

Integral Domains in Which Every Nonzero w-Flat Ideal Is w-Invertible

Let D be an integral domain and w be the so-called w-operation on D. We define D to be a w-FF domain if every w-flat w-ideal of D is of w-finite type. This paper presents some properties of w-FF domains and related domains. Among other things, we study the w-FF property in the polynomial extension, the t-Nagata ring and the pullback construction.

When an Extension of Nagata Rings Has Only Finitely Many Intermediate Rings, Each of Those Is a Nagata Ring

LetR⊂Sbe an extension of commutative rings, withXan indeterminate, such that the extensionRX⊂SXof Nagata rings has FIP (i.e.,SXhas only finitely manyRX-subalgebras). Then, the number ofRX-subalgebras ofSXequals the number ofR-subalgebras ofS. In fact, the function from the set ofR-subalgebras ofSto the set ofRX-subalgebras ofSXgiven byT ↦TXis an order-isomorphism.

Asymptotic Behavior of Integral Closures in Modules

Let R be a commutative Noetherian Nagata ring, let M be a non-zero finitely generated R-module, and let I be an ideal of R such that height MI > 0. In this paper, there is a definition of the integral closure Na for any submodule N of M extending Rees' definition for the case of a domain. As the main results, it is shown that the operation N → Na on the set of submodules N of M is a semi-prime operation, and for any submodule N of M, the sequences Ass R M/(InN)a and Ass R (InM)a/(InN)a(n=1,2,…) of associated prime ideals are increasing and ultimately constant for large n.

Nagata rings and directed unions of Artinian subrings

We investigate when a Nagata ringR(X)can be written as a directed union of Artinian subrings. For a family of zero-dimensional rings{Rα}α∈A, we show that∏α∈ARα(Xα)is not a directed sum of Artinian subrings.