arXiv subjects

Ryota Kuroki

Publications and source records attributed to Ryota Kuroki.

4 recordsLinked to original sources

arXiv2025

A quantitative Hilbert's basis theorem and the constructive Krull dimension

In classical mathematics, Gulliksen has introduced the length of Noetherian modules, and Brookfield has determined the length of Noetherian polynomial rings. Brookfield's result can be regarded as a quantitative version of Hilbert's basis theorem. In this paper, based on the inductive definition of Noetherian modules in constructive algebra, we introduce a constructive version of the length called $α$-Noetherian modules, and present a constructive proof of some results by Brookfield. As a consequence, we obtain a new constructive proof of $\dim K[X_0,\ldots,X_{n-1}]<1+n$ and $\dim\mathbb{Z}[X_0,\ldots,X_{n-1}]<2+n$, where $K$ is a discrete field.

math.RA↗

arXiv2025

A constructive proof of the general Nullstellensatz for Jacobson rings

We give a constructive proof of the general Nullstellensatz: a univariate polynomial ring over a commutative Jacobson ring is Jacobson. This theorem implies that every finitely generated algebra over a zero-dimensional ring or the ring of integers is Jacobson, which has been an open problem in constructive algebra. We also prove a variant of the general Nullstellensatz for finitely Jacobson rings.

math.AC↗

arXiv2025

A quantitative general Nullstellensatz for Jacobson rings

The general Nullstellensatz states that if $A$ is a Jacobson ring, $A[X]$ is Jacobson. We introduce the notion of an $α$-Jacobson ring for an ordinal $α$ and prove a quantitative version of the general Nullstellensatz: if $A$ is an $α$-Jacobson ring, $A[X]$ is $(α+1)$-Jacobson. The quantitative general Nullstellensatz implies that $K[X_1,\ldots,X_n]$ is not only Jacobson but also $(1+n)$-Jacobson for any field $K$. It also implies that $\mathbb{Z}[X_1,\ldots,X_n]$ is $(2+n)$-Jacobson.

math.AC↗

arXiv2024

A constructive counterpart of the subdirect representation theorem for reduced rings

We give a constructive counterpart of the theorem of Andrunakievič and Rjabuhin, which states that every reduced ring is a subdirect product of domains. As an application, we extract a constructive proof of the fact that every ring $A$ satisfying $\forall x\in A. x^3=x$ is commutative from a classical proof. We also prove a similar result for semiprime ideals.

math.RA↗