Sum of ideals

Definition

The sum of ideals is an operation in ring theory where two or more ideals are combined to form a new ideal that contains all elements that can be expressed as sums of elements from the original ideals. This concept helps to understand how ideals interact within a ring and plays a critical role in various algebraic structures, including varieties and their geometric interpretations.

5 Must Know Facts For Your Next Test

1. The sum of two ideals, I and J, denoted as I + J, consists of all elements of the form a + b, where a ∈ I and b ∈ J.
2. The sum of ideals is itself an ideal; it satisfies the closure properties under addition and absorbs multiplication by ring elements.
3. If I and J are two ideals in a commutative ring, then their sum is the smallest ideal containing both I and J.
4. The sum of ideals can be extended to any finite collection of ideals, not just two, by adding the corresponding generators.
5. In the context of varieties, the sum of ideals reflects the union of their corresponding algebraic sets, highlighting their geometrical relationships.

Review Questions

• How does the sum of ideals relate to the operations on varieties in algebraic geometry?
  • The sum of ideals directly influences the geometric interpretation in algebraic geometry since it corresponds to the union of the varieties defined by those ideals. When we take the sum of two ideals, the resulting ideal represents points that belong to either variety or both. This connection allows us to analyze how different algebraic structures coexist and interact on a geometric level.
• Discuss how the sum of two distinct ideals might affect their intersection and what implications this has for understanding their properties.
  • When summing two distinct ideals, the resulting ideal may have a different intersection with other ideals than either original ideal alone. For instance, if I and J have non-trivial intersections, then I + J could capture additional relationships between elements beyond what each ideal represents separately. Understanding these interactions is crucial for exploring deeper properties such as maximality or primality in algebraic structures.
• Evaluate the significance of the sum of ideals in defining new algebraic structures within ring theory, particularly in relation to generated ideals.
  • The sum of ideals plays a crucial role in defining new algebraic structures within ring theory, especially regarding generated ideals. When we consider a collection of ideals and form their sum, we effectively generate a new ideal that encompasses all their contributions. This process not only enriches our understanding of existing ideals but also allows for the exploration of properties like closure and minimality, leading to further developments in both algebraic theory and applications in geometry.

Related terms

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

intersection of ideals: The set of elements that are common to two or more ideals, forming another ideal that captures shared properties.

generated ideal: An ideal formed by taking all possible finite sums of products of elements from a given set with elements from the ring.