### Article

Kyungpook Mathematical Journal 2023; 63(1): 11-13

**Published online** March 31, 2023

Copyright © Kyungpook Mathematical Journal.

### Noether Normalization Implies Full Form of Hilbert Nullstellensatz Theorem

Alborz Azarang

Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz-Iran

e-mail : a_azarang@scu.ac.ir

**Received**: February 24, 2022; **Accepted**: July 25, 2022

We give a new proof for the full form of Hilbert’s Nullstellensatz based on on integral extension and Noether’s Normalization Lemma.

**Keywords**: Field, Noether Normalization

### 1. Introduction

Recall the statements of the the weak and strong forms Hilberts's Nullstellensatz and of Noether's Nomalisation Lemma.

**Theorem 1.1. (Hilbert's Nullstellensatz (weak form)** Let

**Theorem 1.2. (Hilbert's Nullstellensatz (full form)** Let

**Theorem 1.3.** (**Noether's Normalization Lemma**) Let

Usually, the weak form of Hilbert's Nullstellensatz is proved by some technical and classical facts such as Noether's Normalization Lemma, Zariski's Lemma, Artin-Tate Lemma and the concept of

In this short note we prove that Noether's Normalization Lemma also implies the full form of Hilbert's Nullstellensatz Theorem.

Let us recall some standard definitions and facts from commutative ring theory which will be used in this note. For a ring

**Hilbert's Nullstellensatz (full form)**. Note that it is clear that if there exists a natural number

This work was supported by Research Council of Shahid Chamran University of Ahvaz (Ahvaz-Iran) Grant Number: SCU.MM1400.721.

- A. Azarang,
A simple proof of Zariski's lemma , Bull. Iranian Math. Soc.,43(5) (2017), 1529-1530. - P. M. Cohn. Basic Algebra: Groups. Rings and Fields. London: Springer-Verlag London; 2003.
- J. A. Eagon,
Finitely generated domain over Jacobson semisimple rings are Jacobson semisimple , Amer. Math. Monthly,74 (1967), 1091-1092. - D. Eisenbud. Commutative Algebra: With a View Toward Algebraic Geometry. Graduate Texts in Mathematics. New York: Springer-Verlag; 1995.
- R. Gilmer. Multiplicative Ideal Theory. Pure and Applied Mathematics. New York: Marcel-Dekker; 1972.
- I. Kaplansky. Commutative Rings. Chicago: The University of Chicago Press; 1974.
- R. Y. Sharp. Steps in Commutative Algebra. London Mathemathecal Society Student Texts. Camberidge: Camberidge University Press; 2000.