Nagata Ring Research Articles

On generalized Star-GCD extensions

– Let ⋆ be a star operation on a ring extension R ⊆ S . The ring extension is said to be a generalized ⋆ -greatest common divisor (G- ⋆ GCD) extension if the intersection of two ⋆ -invertible ideals of R is ⋆ -invertible. We investigate properties of G- ⋆ GCD extensions. We give special attention to polynomial rings and Nagata rings.

Non-integrally closed Kronecker function rings and integral domains with a unique minimal overring

It is well-known that an integrally closed domain D can be expressed as the intersection of its valuation overrings but, if D is not a Prüfer domain, most of the valuation overrings of D cannot be seen as localizations of D. The Kronecker function ring of D is a classical construction of a Prüfer domain which is an overring of D[t], and its localizations at prime ideals are of the form V(t) where V runs through the valuation overrings of D. This fact can be generalized to arbitrary integral domains by expressing them as intersections of overrings which admit a unique minimal overring. In this article we first continue the study of rings admitting a unique minimal overring extending known results obtained in the 1970s and constructing examples where the integral closure is very far from being a valuation domain. Then we extend the definition of Kronecker function ring to the non-integrally closed setting by studying intersections of Nagata rings of the form A(t) for A an integral domain admitting a unique minimal overring.

Quasi-S-primary ideals of commutative rings

Let R be a commutative ring with and S be a multiplicatively closed subset of R. We call an ideal I of R disjoint with S quasi-S-primary if there exists an such that whenever and then or . We investigate many properties and characterizations of quasi-S-primary ideals. We discuss the form of quasi-S-primary ideals in polynomial, power series, the Serre’s conjecture and the Nagata rings. Futhermore, we study quasi-S-primary ideals in amalgamated algebras. Our results allow us to construct original examples of quasi-S-primary ideals.

Some results on t–locally Strong Mori domains and their rings of integer–valued polynomials

An integral domain D is said to be t–locally Strong Mori (for short, t–LSM) if $$D_\mathfrak {m}$$ is Strong Mori for all t–maximal ideals $$\mathfrak {m}$$ of D. This paper studies some ring–theoretic properties of t–LSM domains and the algebra structure of rings of integer–valued polynomials arising from t–LSM domains. Among other things, we investigate the property of being a t–LSM domain in the t–flat overring extension, the t–Nagata ring, the polynomial ring, pullback construction and the power series ring. Also, we study $${\mathrm {Int}}(D)$$ over a t–LSM domain D. Precisely, we are interested in the Krull dimension, the trivial case and some module structure properties.

Semistar ascending chain conditions over polynomial rings

Let D be an integral domain. G. Picozza associated to a stable semistar operation $$\star $$ on D, a semistar operation $$\star _1$$ on the polynomial ring D[X]. We defined the notion of semistar accr domain. We prove that D is $${\widetilde{\star }}$$ -Noetherian if and only D[X] is $$\star _1$$ -accr. On the other hand, we prove that D satisfies the ascending chain condition on radical quasi- $${\widetilde{\star }}$$ -ideals if and only if D[X] satisfies the ascending chain condition on radical quasi- $$\star _1$$ -ideals if and only if the Nagata ring of D with respect to the semistar operation $${\widetilde{\star }}$$ satisfies the ascending chain condition on radical ideals.

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.

Trace properties and the rings $$R({\scriptstyle {\mathrm{X}}})$$ and $$R\langle {\scriptstyle {\mathrm{X}}}\rangle $$

An integral domain R is an RTP domain (or has the radical trace property) (resp. an LTP domain), if I(R : I) is a radical ideal for each nonzero noninvertible ideal I (resp. $$I(R:I)R_P=PR_P$$ for each minimal prime P of I(R : I)). Clearly each RTP domain is an LTP domain, but whether the two are equivalent is open except in certain special cases. In this paper, we study when the Nagata ring $$R({\scriptstyle {\mathrm{X}}})$$ and the ring $$R\langle {\scriptstyle {\mathrm{X}}}\rangle $$ are LTP (resp. RTP) domains in different contexts of integral domains such as integrally closed domains, Noetherian and Mori domains, pseudo-valuation domains and more. We also study the descent of these notions from particular overrings of R to R itself.

Radical factorization in finitary ideal systems

In this article, we investigate the concept of radical factorization with respect to finitary ideal systems of cancellative monoids. We present new characterizations for r-almost Dedekind r-SP-monoids and provide specific descriptions of t-almost Dedekind t-SP-monoids and w-SP-monoids. We show that a monoid is a w-SP-monoid if and only if the radical of every nontrivial principal ideal is t-invertible. We characterize when the monoid ring is a w-SP-domain and describe when the *-Nagata ring is an SP-domain for a star operation * of finite type.

Two applications of Nagata rings and modules

Let [Formula: see text] be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China 10 (2015) 91–110]. First they are used to obtain Chase’s Theorem for [Formula: see text]-coherent rings. In particular, we obtain the [Formula: see text]-version of Chase’s Theorem, where [Formula: see text] is the classical star operation in ideal theory. In the second half, we apply they to characterize [Formula: see text]-flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)].

Sharpness and semistar operations in Prüfer-like domains

Let be a semistar operation on a domain D, the finite-type semistar operation associated to , and D a Prüfer -multiplication domain (PMD). For the special case of a Prüfer domain (where is equal to the identity semistar operation), we show that a nonzero prime P of D is sharp, that is, that , where the intersection is taken over the maximal ideals M of D that do not contain P, if and only if two closely related spectral semistar operations on D differ. We then give an appropriate definition of -sharpness for an arbitrary PMD D and show that a nonzero prime P of D is -sharp if and only if its extension to the -Nagata ring of D is sharp. Calling a PMD -sharp (-doublesharp) if each maximal (prime) -ideal of D is sharp, we also prove that such a D is -doublesharp if and only if each -linked overring of D is -sharp.

Semistar-super potent domains

Let \(\star \) be a finite type star operation defined on an integral domain D. Call D\(\star \)-potent (resp., \(\star \)-super potent) if every maximal \(\star \)-ideal M of D containes a nonzero finitely generated ideal I such that I is contained in no other maximal \(\star \)-ideal of D (resp., I is contained in no other maximal \(\star \)-ideal and every finitely generated ideal J containing I is \(\star \)-invertible). In this paper, we generalize the notion of super potent domains in the context of semistar operations. We prove that if \(\star \) is a semistar operation on an integral domain D, then D is \(\widetilde{\star }\)-(super) potent if and only if D[X] is \([\star ]\)-(super) potent if and only if the Nagata ring associated to \(\star \) is (super) potent, where \([\star ]\) is the semistar operation introduced by Chang and Fontana (J Algebra 318(1):484–493, 2007).

Prüfer conditions in the Nagata ring and the Serre’s conjecture ring

ABSTRACTThe Nagata ring R(X) and the Serre’s conjecture ring R⟨X⟩ are two localizations of the polynomial ring R[X] at the polynomials of unit content and at the monic polynomials, respectively. In this paper, we contribute to the study of Prüfer conditions in R(X) and R⟨X⟩. In particular, we solve the four open questions posed by Glaz and Schwarz in Section 8 of their survey paper [38] related to the transfer of Prüfer conditions to these two constructions.

Graded integral domains and Nagata rings, II

Let $D$ be an integral domain with quotient field $K$, $X$ be an indeterminate over $D$, $K[X]$ be the polynomial ring over $K$, and $R= \{f \in K[X] \mid f(0) \in D\}$; so $R$ is a subring of $K[X]$ containing $D[X]$. For $f = a_0 + a_1X + \cdots + a_nX^n \in R$, let $C(f)$ be the ideal of $R$ generated by $a_0, a_1X, \dots , a_nX^n$ and $N(H) = \{g \in R \mid C(g)_v = R\}$. In this paper, we study two rings $R_{N(H)}$ and Kr$(R, v) = \{\frac{f}{g} \mid f, g \in R$, $g \neq 0$, and $C(f) \subseteq C(g)_v\}$. We then use these two rings to give some examples which show that the results of \cite{ac13} are the best generalizations of Nagata rings and Kronecker function rings to graded integral domains.

WHEN THE NAGATA RING D(X) IS A SHARP DOMAIN

Let $D$ be an integral domain, $X$ be an indeterminate over $D$, $D[X]$ be the polynomial ring over $D$, and $D(X)$ be the Nagata ring of $D$. Let $[d]$ be the star operation on $D[X]$, which is an extension of the $d$-operation on $D$ as in [5,Theorem 2.3]. In this paper, we show that $D$ is a sharp domain if and only if $D[X]$ is a $[d]$-sharp domain, if and only if $D(X)$ is a sharp domain.

Power series over Noetherian domains, Nagata rings, and Kronecker function rings

Let D be a Noetherian domain, ⁎ be a star operation on D, X be an indeterminate over D, D[[X]] be the power series ring over D, c(f) be the ideal of D generated by the coefficients of f∈D[[X]], and N⁎={f∈D[[X]]|c(f)⁎=D}. Moreover, if ⁎ is e.a.b., then we let Kr((D,⁎))={fg|f,g∈D[[X]],0≠g, and c(f)⊆c(g)⁎}. In this paper, we show that N⁎ is a saturated multiplicative set and Kr((D,⁎)) is a Bezout domain. We then study some ring-theoretic properties of D[[X]]N⁎ and Kr((D,⁎)). For example, we prove that every invertible ideal of D[[X]]N⁎ is principal; dim(Kr((D,b)))=dimv(D); and if V is a valuation overring of D, then Vˆ={fg|f,g∈D[[X]],g≠0, and c(f)V⊆c(g)V} is a valuation overring of D[[X]]; and Kr((D,⁎))=⋂{Vˆ|V is a ⁎-valuation overring of D}.

ON PIECEWISE NOETHERIAN DOMAINS

In this paper, we study piecewise Noetherian (resp., piecewise w-Noetherian) properties in several settings including flat (resp., t-flat) overrings, Nagata rings, integral domains of finite character (resp., w-finite character), pullbacks of a certain type, polynomial rings, and D + XK[X] constructions.

A Note on S-Noetherian Domains

Let D be an integral domain, t be the so-called t-operation on D; and S be a (not necessarily saturated) multiplicative subset of D. In this paper, we study the Nagata ring of S-Noetherian domains and locally S-Noetherian domains. We also inves- tigate the t-Nagata ring of t-locally S-Noetherian domains. In fact, we show that if S is an anti-archimedean subset of D, then D is an S-Noetherian domain (respectively, locally S-Noetherian domain) if and only if the Nagata ring D(X)N is an S-Noetherian domain (respectively, locally S-Noetherian domain). We also prove that if S is an anti-archimedean subset of D, then D is a t-locally S-Noetherian domain if and only if the polynomial ring D(X) is a t-locally S-Noetherian domain, if and only if the t-Nagata ring D(X)Nv is a t-locally S-Noetherian domain.

Commutative weakly nil clean unital rings

We define the concept of a weakly nil clean commutative ring which generalizes Diesel's [11] notion of a nil clean commutative ring, and investigate this class of rings. We obtain some fundamental properties. In particular, it is proved that these rings are clean. We also consider the questions of when the Nagata ring as well as the group ring is weakly nil clean.

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.

Graded integral domains and Nagata rings

Let R=⨁α∈ΓRα be an integral domain graded by an arbitrary torsionless grading monoid Γ. For any f∈R, let C(f) be the ideal of R generated by the homogeneous components of f, and let N(H)={g∈R|C(g)v=R}. In this paper, we study relationships between the ideal-theoretic properties of RN(H) and the homogeneous ideal-theoretic properties of R. For example, we show that R is a graded Krull domain if and only if RN(H) is a Dedekind domain, if and only if RN(H) is a PID; and that if R contains a unit of nonzero degree, then R is a PvMD if and only if RN(H) is a Prüfer domain, if and only if each ideal of RN(H) is extended from a homogeneous ideal of R.