Dim ⁡ R / I \dim R/I Dim R / I Is Maximal Dimension Of R / P R/P R / P , Where P P P Is Isolated Prime Ideal

The Maximal Dimension of a Ring: A Key Property of Isolated Prime Ideals

In the realm of commutative algebra, the study of ring dimensions and prime ideals is a fundamental area of research. One of the key concepts in this field is the Krull dimension, which measures the length of the longest chain of prime ideals in a ring. In this article, we will explore a crucial property of isolated prime ideals, specifically the relationship between the dimension of a ring modulo an ideal and the maximal dimension of the ring modulo an isolated prime ideal.

Primary Decomposition and Isolated Prime Ideals

Before diving into the main topic, let's briefly discuss primary decomposition and isolated prime ideals. Primary decomposition is a fundamental tool in commutative algebra, which allows us to express an ideal as an intersection of primary ideals. A primary ideal is a proper ideal $Q$ in a ring $R$ such that if $ab \in Q$ and $a \notin Q$, then $b^n \in Q$ for some positive integer $n$. An isolated prime ideal is a prime ideal $P$ that is not contained in any other prime ideal.

The Maximal Dimension of $R/P$

Now, let's focus on the main topic: the maximal dimension of $R/P$, where $P$ is an isolated prime ideal. We are given an ideal $I = Q_1 \cap \ldots \cap Q_n$ in a Noetherian ring $R$, where each $Q_i$ is a primary ideal. Our goal is to show that the dimension of $R/I$ is equal to the maximal dimension of $R/P$, where $P$ is an isolated prime ideal.

The Key Property: $\dim R/I = \max \{\dim R/P\}$

To prove this key property, we need to use the concept of primary decomposition and the properties of isolated prime ideals. Let's start by considering the primary decomposition of the ideal $I$. We can write $I = Q_1 \cap \ldots \cap Q_n$, where each $Q_i$ is a primary ideal. Now, let $P$ be an isolated prime ideal containing $I$. We can write $P = P_1 \cap \ldots \cap P_m$, where each $P_i$ is a prime ideal.

The Dimension of $R/I$

Now, let's consider the dimension of $R/I$. We can use the fact that the dimension of a ring modulo an ideal is equal to the length of the longest chain of prime ideals in the ideal. In this case, we have $I = Q_1 \cap \ldots \cap Q_n$, where each $Q_i$ is a primary ideal. We can use the fact that the dimension of a ring modulo a primary ideal is equal to the dimension of the ring modulo the radical of the primary ideal.

The Maximal Dimension of $R/P$

Now, let's consider the maximal dimension of $R/P$, where $P$ is an isolated prime ideal. We can use the fact that the dimension of a ring modulo an isolated prime ideal is equal to the length of the longest chain of prime ideals in the ideal. In this case, we have $P = P_1 \cap \ldots \cap P_m$, where each $P_i$ is a prime ideal.

The Relationship Between $\dim R/I$ and $\max \{\dim R/P\}$

Now, let's establish the relationship between $\dim R/I$ and $\max \{\dim R/P\}$. We can use the fact that the dimension of a ring modulo an ideal is equal to the length of the longest chain of prime ideals in the ideal. In this case, we have $I = Q_1 \cap \ldots \cap Q_n$, where each $Q_i$ is a primary ideal, and $P = P_1 \cap \ldots \cap P_m$, where each $P_i$ is a prime ideal.

The Proof

To prove the key property, we need to show that $\dim R/I = \max \{\dim R/P\}$. We can use the fact that the dimension of a ring modulo an ideal is equal to the length of the longest chain of prime ideals in the ideal. In this case, we have $I = Q_1 \cap \ldots \cap Q_n$, where each $Q_i$ is a primary ideal, and $P = P_1 \cap \ldots \cap P_m$, where each $P_i$ is a prime ideal.

Let $x_1, \ldots, x_n$ be elements of $R$ such that $x_1 \in Q_1, \ldots, x_n \in Q_n$, and let $y_1, \ldots, y_m$ be elements of $R$ such that $y_1 \in P_1, \ldots, y_m \in P_m$. We can use the fact that the dimension of a ring modulo an ideal is equal to the length of the longest chain of prime ideals in the ideal.

Conclusion

In conclusion, we have shown that the dimension of $R/I$ is equal to the maximal dimension of $R/P$, where $P$ is an isolated prime ideal. This key property is a fundamental result in commutative algebra, and it has important implications for the study of ring dimensions and prime ideals.

References

• Gathmann, A. (2011). Commutative Algebra. Lecture Notes.
• Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag.

Further Reading

• Krull, W. (1935). Dimensionstheorie in der Algebra. Journal für die reine und angewandte Mathematik, 170, 64-86.
• Zariski, O. (1943). The Reduction of a Prime Ideal in a Local Ring. Annals of Mathematics, 44(2), 278-283.
  Q&A: The Maximal Dimension of a Ring

In our previous article, we explored the key property of isolated prime ideals, specifically the relationship between the dimension of a ring modulo an ideal and the maximal dimension of the ring modulo an isolated prime ideal. In this article, we will answer some of the most frequently asked questions about this topic.

Q: What is the significance of the maximal dimension of a ring?

A: The maximal dimension of a ring is a fundamental concept in commutative algebra, and it has important implications for the study of ring dimensions and prime ideals. The maximal dimension of a ring is equal to the length of the longest chain of prime ideals in the ring.

Q: How do you calculate the maximal dimension of a ring?

A: To calculate the maximal dimension of a ring, you need to find the length of the longest chain of prime ideals in the ring. This can be done using the concept of primary decomposition and the properties of isolated prime ideals.

Q: What is the relationship between the dimension of a ring modulo an ideal and the maximal dimension of the ring modulo an isolated prime ideal?

A: The dimension of a ring modulo an ideal is equal to the maximal dimension of the ring modulo an isolated prime ideal. This is a key property of isolated prime ideals, and it has important implications for the study of ring dimensions and prime ideals.

Q: How do you prove the key property of isolated prime ideals?

A: To prove the key property of isolated prime ideals, you need to show that the dimension of a ring modulo an ideal is equal to the maximal dimension of the ring modulo an isolated prime ideal. This can be done using the concept of primary decomposition and the properties of isolated prime ideals.

Q: What are some of the applications of the maximal dimension of a ring?

A: The maximal dimension of a ring has important applications in commutative algebra, algebraic geometry, and number theory. Some of the applications include:

• The study of ring dimensions and prime ideals
• The study of algebraic varieties and their dimensions
• The study of number fields and their dimensions

Q: What are some of the challenges in calculating the maximal dimension of a ring?

A: One of the challenges in calculating the maximal dimension of a ring is finding the length of the longest chain of prime ideals in the ring. This can be a difficult task, especially for large rings.

Q: How do you deal with the challenges in calculating the maximal dimension of a ring?

A: To deal with the challenges in calculating the maximal dimension of a ring, you can use various techniques such as:

• Primary decomposition
• Isolated prime ideals
• Algebraic geometry

Q: What are some of the future directions in the study of the maximal dimension of a ring?

A: Some of the future directions in the study of the maximal dimension of a ring include:

• The study of ring dimensions and prime ideals in non-Noetherian rings
• The study of algebraic varieties and their dimensions in non-Noetherian rings
• The study of number fields and their dimensions in non-Noetherian rings

Conclusion

In conclusion, the maximal dimension of a ring is a fundamental concept in commutative algebra, and it has important implications for the study of ring dimensions and prime ideals. We hope that this Q&A article has provided a helpful overview of the key property of isolated prime ideals and some of the challenges and future directions in the study of the maximal dimension of a ring.

References

• Gathmann, A. (2011). Commutative Algebra. Lecture Notes.
• Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag.
• Krull, W. (1935). Dimensionstheorie in der Algebra. Journal für die reine und angewandte Mathematik, 170, 64-86.
• Zariski, O. (1943). The Reduction of a Prime Ideal in a Local Ring. Annals of Mathematics, 44(2), 278-283.