Dimension of a ring

Definition

The dimension of a ring refers to the Krull dimension, which is the supremum of the lengths of all chains of prime ideals in that ring. This concept provides insight into the structure and properties of the ring by examining how prime ideals can be organized. A higher dimension indicates a richer structure in terms of the relationships between its prime ideals, which can influence other characteristics such as the behavior of modules over the ring and its algebraic geometry aspects.

5 Must Know Facts For Your Next Test

1. The Krull dimension of a Noetherian ring is finite and can be determined by analyzing its prime ideals.
2. If a ring has Krull dimension 0, it implies that every prime ideal is maximal, indicating a particular simplicity in its structure.
3. In a unique factorization domain (UFD), the Krull dimension equals one because every non-zero prime ideal can be generated by an irreducible element.
4. The dimension can affect various properties like regularity, Cohen-Macaulayness, and singularities in algebraic geometry.
5. For polynomial rings over a field, the dimension is equal to the number of variables, which reflects their geometric interpretation.

Review Questions

• How does the concept of Krull dimension help in understanding the structure of a ring?
  • Krull dimension serves as a crucial tool in analyzing the structure of a ring by looking at chains of prime ideals. By determining the longest chain, one can gain insights into how complex or simple the ring's structure is. For example, a higher Krull dimension indicates more intricate relationships between prime ideals, which can influence other properties such as module behavior and geometric aspects.
• Discuss the implications of having a Krull dimension equal to zero in a commutative ring.
  • If a commutative ring has a Krull dimension equal to zero, it means that all its prime ideals are maximal. This situation implies that there is no room for non-maximal prime ideals, leading to a straightforward structure where every ideal is essentially 'complete' within its own context. This characteristic simplifies many operations within the ring and indicates that it may behave similarly to fields or finite-dimensional spaces.
• Evaluate how changing the dimension of a polynomial ring influences its geometric interpretations.
  • Changing the dimension of a polynomial ring directly influences its geometric interpretations by altering the number of variables involved. In particular, for a polynomial ring in $n$ variables over a field, its Krull dimension is $n$. This means that geometrically, it corresponds to an $n$-dimensional affine space. If additional variables are added or removed, this changes not just the dimensionality but also potentially affects aspects like singularities and intersections when considering varieties defined by these polynomials.

Related terms

Prime ideal: A prime ideal is an ideal in a commutative ring such that if the product of two elements is in the ideal, at least one of those elements must also be in the ideal.

Chain of prime ideals: A chain of prime ideals is a sequence of prime ideals where each prime ideal is contained in the next, illustrating a specific order and hierarchy among them.

Height of a prime ideal: The height of a prime ideal is defined as the length of the longest chain of prime ideals contained within it, giving an idea about its position in the spectrum of the ring.