Converse To Hilbert Basis Theorem

Introduction

The Hilbert Basis Theorem is a fundamental result in commutative algebra, which states that if $R$ is a Noetherian ring, then the polynomial ring $R[x]$ is also Noetherian. In this article, we will explore the converse of this theorem, which is a more challenging result. We will prove that if the polynomial ring $R[x]$ is Noetherian, then $R$ is also Noetherian.

Background

To understand the converse of the Hilbert Basis Theorem, we need to recall some basic definitions and results from commutative algebra.

• A ring $R$ is said to be Noetherian if every non-empty set of ideals in $R$ has a maximal element. Equivalently, $R$ is Noetherian if every ideal in $R$ is finitely generated.
• The polynomial ring $R[x]$ is the ring of polynomials with coefficients in $R$. A polynomial in $R[x]$ is a formal expression of the form $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where $a_i \in R$ for each $i$.
• The Hilbert Basis Theorem states that if $R$ is Noetherian, then $R[x]$ is also Noetherian.

Proof of the Converse

To prove the converse of the Hilbert Basis Theorem, we will use a proof by contradiction. Assume that $R$ is not Noetherian, and let $I$ be a non-finitely generated ideal in $R$. We will show that this assumption leads to a contradiction.

Step 1: Constructing a Non-Finitely Generated Ideal in $R[x]$

Let $I$ be a non-finitely generated ideal in $R$. We will construct a non-finitely generated ideal in $R[x]$ using the elements of $I$. Let $f_1, f_2, \ldots$ be a sequence of elements in $I$ such that no finite subset of $\{f_1, f_2, \ldots\}$ generates $I$. We will show that the ideal $J$ generated by $\{f_1, f_2, \ldots\}$ in $R[x]$ is also non-finitely generated.

Step 2: Showing that $J$ is Non-Finitely Generated

Suppose, for the sake of contradiction, that $J$ is finitely generated. Then there exist polynomials $g_1, g_2, \ldots, g_n \in R[x]$ such that $J = \langle g_1, g_2, \ldots, g_n \rangle$. Since each $g_i$ is a polynomial, it has a finite number of terms. Let $m$ be the maximum degree of the terms in the polynomials $g_1, g_2, \ldots, g_n$. Then each term in $J$ has degree at most $m$.

Step 3: Deriving a Contradiction

Since $I$ is non-finitely generated, there exists an element $f \in I$ such that no finite subset of $\{f_1, f_2, \ldots\}$ generates $I$. We will show that this element $f$ leads to a contradiction.

Since $f \in I$, we have $f \in J$. Therefore, $f$ can be written as a linear combination of the generators $g_1, g_2, \ldots, g_n$ of $J$. Let $f = a_1 g_1 + a_2 g_2 + \cdots + a_n g_n$, where $a_i \in R[x]$ for each $i$. Since each $g_i$ has degree at most $m$, we have $\deg f \leq m$. However, since $f \in I$, we have $\deg f \geq m+1$. This is a contradiction, since $\deg f$ cannot be both less than or equal to $m$ and greater than or equal to $m+1$.

Conclusion

We have shown that if $R$ is not Noetherian, then $R[x]$ is also not Noetherian. This completes the proof of the converse of the Hilbert Basis Theorem.

Implications

The converse of the Hilbert Basis Theorem has important implications for the study of commutative algebra. It shows that the Noetherian property of a ring is preserved under the operation of taking polynomial rings. This result has far-reaching consequences for the study of algebraic geometry, algebraic number theory, and other areas of mathematics.

References

• Hilbert, D. (1890). "Über die Theorie der algebraischen Formen." Mathematische Annalen, 36(4), 473-534.
• Zariski, O., & Samuel, P. (1958). Commutative Algebra. Vol. 1. Van Nostrand.

Further Reading

For further reading on the Hilbert Basis Theorem and its converse, we recommend the following texts:

• Atiyah, M. F., & Macdonald, I. G. (1969). Introduction to Commutative Algebra. Addison-Wesley.
• Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag.

Introduction

In our previous article, we proved the converse of the Hilbert Basis Theorem, which states that if the polynomial ring $R[x]$ is Noetherian, then $R$ is also Noetherian. In this article, we will answer some frequently asked questions about the converse of the Hilbert Basis Theorem.

Q: What is the significance of the converse of the Hilbert Basis Theorem?

A: The converse of the Hilbert Basis Theorem is significant because it shows that the Noetherian property of a ring is preserved under the operation of taking polynomial rings. This result has far-reaching consequences for the study of algebraic geometry, algebraic number theory, and other areas of mathematics.

Q: How does the converse of the Hilbert Basis Theorem relate to the Hilbert Basis Theorem?

A: The converse of the Hilbert Basis Theorem is the opposite of the Hilbert Basis Theorem. While the Hilbert Basis Theorem states that if $R$ is Noetherian, then $R[x]$ is also Noetherian, the converse states that if $R[x]$ is Noetherian, then $R$ is also Noetherian.

Q: What are some examples of rings that are Noetherian?

A: Some examples of rings that are Noetherian include:

• The ring of integers, $\mathbb{Z}$
• The ring of polynomials over a field, $F[x]$
• The ring of polynomials over a Noetherian ring, $R[x]$

Q: What are some examples of rings that are not Noetherian?

A: Some examples of rings that are not Noetherian include:

• The ring of rational numbers, $\mathbb{Q}$
• The ring of real numbers, $\mathbb{R}$
• The ring of polynomials over a non-Noetherian ring, $R[x]$ where $R$ is not Noetherian

Q: How does the converse of the Hilbert Basis Theorem relate to the concept of Noetherian rings?

A: The converse of the Hilbert Basis Theorem is a fundamental result in the study of Noetherian rings. It shows that the Noetherian property of a ring is preserved under the operation of taking polynomial rings, which is a key concept in the study of Noetherian rings.

Q: What are some applications of the converse of the Hilbert Basis Theorem?

A: The converse of the Hilbert Basis Theorem has far-reaching consequences for the study of algebraic geometry, algebraic number theory, and other areas of mathematics. Some applications of the converse of the Hilbert Basis Theorem include:

• The study of algebraic curves and surfaces
• The study of algebraic number fields
• The study of commutative algebra and ring theory

Q: What are some open problems related to the converse of the Hilbert Basis Theorem?

A: Some open problems related to the converse of the Hilbert Basis Theorem include:

• The study of the converse of the Hilbert Basis Theorem for non-commutative rings
• The study of the converse of the Hilbert Basis Theorem for rings with zero divisors
• The study of the converse of the Hilbert Basis Theorem for rings with non-trivial nilpotent elements

Conclusion

We hope this article has provided a clear and concise answer to some frequently asked questions about the converse of the Hilbert Basis Theorem. We welcome any feedback or comments on this article.

References

• Hilbert, D. (1890). "Über die Theorie der algebraischen Formen." Mathematische Annalen, 36(4), 473-534.
• Zariski, O., & Samuel, P. (1958). Commutative Algebra. Vol. 1. Van Nostrand.

Further Reading

For further reading on the converse of the Hilbert Basis Theorem, we recommend the following texts:

• Atiyah, M. F., & Macdonald, I. G. (1969). Introduction to Commutative Algebra. Addison-Wesley.
• Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag.