Maximal ideal

Definition

A maximal ideal is an ideal in a ring that is not equal to the entire ring and is maximal with respect to the property of being an ideal. This means that if you have an ideal that properly contains it, that ideal must be the entire ring itself. Maximal ideals are essential in understanding the structure of rings, particularly when examining quotient rings and their properties.

5 Must Know Facts For Your Next Test

1. Every maximal ideal is a prime ideal, but not every prime ideal is maximal.
2. The quotient ring formed by a ring and a maximal ideal is always a field.
3. Maximal ideals correspond to points in the spectrum of a ring, revealing important information about its structure.
4. In a finite-dimensional vector space over a field, every non-zero subspace is contained in a maximal subspace, mirroring the idea of maximal ideals in rings.
5. In commutative rings, the Jacobson radical consists of all elements contained in every maximal ideal.

Review Questions

• How does the concept of maximal ideals relate to the structure of quotient rings?
  • Maximal ideals play a crucial role in the formation of quotient rings. When you take a ring and factor it by a maximal ideal, the resulting quotient ring is guaranteed to be a field. This connection highlights the significance of maximal ideals in understanding how rings can be simplified and studied through their quotient structures.
• Compare and contrast maximal ideals with prime ideals, explaining their relationships within a ring.
  • Maximal ideals and prime ideals are related concepts in ring theory. Every maximal ideal is also a prime ideal because if a product of two elements lies within a maximal ideal, at least one element must be there as well. However, not all prime ideals are maximal; there can be prime ideals that are not contained in any maximal ideals, showing different levels of containment within the hierarchy of ideals in a ring.
• Evaluate the implications of maximal ideals on the overall structure of commutative rings and their applications in algebraic geometry.
  • Maximal ideals significantly impact the structure of commutative rings, particularly through their connection to fields when forming quotient rings. This concept extends into algebraic geometry where points on an algebraic variety correspond to maximal ideals in the polynomial ring. Understanding these relationships allows mathematicians to study geometric properties using algebraic methods, highlighting the interplay between algebra and geometry.

Related terms

ideal:

A subset of a ring that absorbs multiplication by elements of the ring and is closed under addition.

quotient ring: A new ring formed by partitioning a given ring into cosets of an ideal.

prime ideal:

An ideal such that if the product of two elements is in the ideal, then at least one of those elements must also be in the ideal.