A generalization of Kronecker function rings and Nagata rings

关于Nagata型多项式自同构

On polynomial automorphisms of Nagata type

本文定义了一族推广Nagata自同构的多项式环同态，给出其成为自同构的充要条件，验证满足Jacobian猜想，并给出野性自同构的必要条件及Łojasiewicz指数上半连续性相关性质。

Characterizations of graded Prüfer $\star$-multiplication domains

Let $R=\bigoplus_{α\inΓ}R_α$ be a graded integral domain graded by an arbitrary grading torsionless monoid $Γ$, and $\star$ be a semistar operation on $R$. In this paper we define and study the graded integral domain analogue of $\star$-Nagata and Kronecker function rings of $R$ with respect to $\star$. We say that $R$ is a graded Prüfer $\star$-multiplication domain if each nonzero finitely generated homogeneous ideal of $R$ is $\star_f$-invertible. Using $\star$-Nagata and Kronecker function rings, we give several different equivalent conditions for $R$ to be a graded Prüfer $\star$-multipli…

纤维积环与Nagata理想化的Hilbert-Kunz重数

Hilbert-Kunz Multiplicity of Fiber Product Rings and Nagata Idealizations

本文为纤维积环和Nagata理想化推导出Hilbert-Kunz重数的显式公式，并给出新界与结构结果。

$(S_2)$-化、半Nagata环与提升问题

$(S_2)$-ifications, semi-Nagata rings, and the lifting problem

该文分两部分研究交换环论问题：一是引入$(S_2)$-化替代正规化研究半Nagata环相关性质，二是证明半局部Noetherian环的多种形式纤维性质可沿理想提升，拓展了环论提升问题的结论。

An affine version of a theorem of Nagata

Let R be an affine k-domain over the field k. The paper's main result is that, if R admits a non-trivial embedding in a polynomial ring K[s] for some field K containing k, then R can be embedded in a polynomial ring F[t] which extends R algebraically. This theorem can be applied to subrings of a ring which admits a non-zero locally nilpotent derivation. In this way, we obtain a concise new proof of the cancellation theorem for rings of transcendence degree one for fields of characteristic zero.

A counterpart to Nagata idealization

Idealization of a module $K$ over a commutative ring $S$ produces a ring having $K$ as an ideal, all of whose elements are nilpotent. We develop a method that under suitable field-theoretic conditions produces from an $S$-module $K$ and derivation $D:S\rightarrow K$ a subring $R$ of $S$ that behaves like the idealization of $K$ but is such that when $S$ is a domain, so is $R$. The ring $S$ is contained in the normalization of $R$ but is finite over $R$ only when $R = S$. We determine conditions under which $R$ is Noetherian, Cohen-Macaulay, Gorenstein, a complete intersection or a hypersurface…

Invariant Rings and Quasiaffine Quotients

We study Hilbert's fourteenth problem from a geometric point of view. Nagata's celebrated counterexample demonstrates that for an arbitrary group action on a variety the ring of invariant functions need not be isomorphic to the ring of functions of an affine variety. Nevertheless one can prove that such a ring of invariants is always isomorphic to the ring of functions on a quasi-affine variety. Conversely, for a given quasi-affine variety V there exists always an action of the additive group on some affine variety W such that the ring of functions of V is isomorphic to the ring of invariant f…

非整闭的 Kronecker 函数环与具有唯一极小扩环的整环

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

本文将 Kronecker 函数环的构造推广到非整闭整环，通过研究具有唯一极小扩环的整环及其 Nagata 环的交集，并构造整闭包远离赋值整环的新例子。

有界余维的完全交：Nagata准则的失效与开性的恢复

Complete Intersections of Bounded Codimension: Failure of the Nagata Criterion and Recovery of Openness

本文证明余维至多为$c$的完全交性质不满足Nagata准则，构造反例并给出NC的刻画条件，恢复其轨迹开性，还证明相关子集在拟紧优概形上可构造。

Cluster Algebras, Invariant Theory, and Kronecker Coefficients I

We relate the $m$-truncated Kronecker products of symmetric functions to the semi-invariant rings of a family of quiver representations. We find cluster algebra structures for these semi-invariant rings when $m=2$. Each {\sf g}-vector cone ${\sf G}_{\Diamond_l}$ of these cluster algebras controls the $2$-truncated Kronecker products for all symmetric functions of degree no greater than $l$. As a consequence, each relevant Kronecker coefficient is the difference of the number of the lattice points inside two rational polytopes. We also give explicit description of all ${\sf G}_{\Diamond_l}$'s. …

A locally F-finite Noetherian domain that is not F-finite

Using an old example of Nagata, we construct a Noetherian ring of prime characteristic p, whose Frobenius morphism is locally finite, but not finite.

准sdf吸收理想在交换环中的研究

Quasi sdf-absorbing ideals in commutative rings

本文研究准sdf吸收理想，探讨其在局部化、满射像等构造下的稳定性，并讨论其素性条件及在整数环中的分类。

A Note on Euclidean Order Types

Euclidean functions with values in an arbitrary well-ordered set were first considered in a 1949 work of Motzkin and studied in more detail in work of Fletcher, Samuel and Nagata in the 1970's and 1980's. Here these results are revisited, simplified, and extended. The two main themes are (i) consideration of Ord-valued functions on an Artinian poset and (ii) use of ordinal arithmetic, including the Hessenberg-Brookfield ordinal sum. In particular, to any Euclidean ring we associate an ordinal invariant, its Euclidean order type, and we initiate a study of this invariant. The main new result gi…

非交换环中几乎素理想与右 $S$-素理想的推广

Generalizations of almost prime and right $S$-prime ideals in noncommutative rings

本文在非交换环中推广了几乎素理想概念，引入了几乎右 $S$-素理想，并利用 Nagata 理想化方法研究其在相关环中的性质与构造。