Ideal Generated by a Set

Definition

An ideal generated by a set is a special subset of a ring that includes all possible finite sums of elements from that set, each multiplied by any element from the ring. This concept is important because it helps in understanding how to form larger structures within rings and is key in defining quotient rings. It also serves as a way to create new ideals and explore properties like maximality and primality within the ring.

5 Must Know Facts For Your Next Test

1. The ideal generated by a set S in a ring R, denoted as (S), consists of all elements of the form r_1s_1 + r_2s_2 + ... + r_ns_n, where r_i are elements from R and s_i are elements from S.
2. Every ideal in a ring can be generated by a single element; such an ideal is called a principal ideal.
3. The intersection of two ideals is also an ideal, and the ideal generated by a set can be viewed as the smallest ideal containing that set.
4. If I is an ideal generated by a set S, then every element in I can be expressed as a linear combination of elements from S using elements from the ring.
5. Ideals generated by different sets may coincide, particularly when one set is a subset of another or when both sets generate the same elements in the ring.

Review Questions

• How does the concept of an ideal generated by a set relate to the structure of rings?
  • The ideal generated by a set establishes a framework for understanding how subsets of rings can be formed into larger structures. By taking all finite combinations of elements from the generating set multiplied by any ring element, it shows how ideals act as building blocks within rings. This understanding is crucial for exploring other concepts like quotient rings, which rely on identifying specific ideals within the ring structure.
• Discuss the differences between principal ideals and ideals generated by multiple elements in terms of their structure and properties.
  • Principal ideals are generated by a single element and have unique properties regarding their simplicity and closure under operations within the ring. In contrast, ideals generated by multiple elements can capture more complex relationships and structures since they include combinations of various elements. While every principal ideal is also an ideal generated by a set, not every ideal generated by a set is principal unless it can be simplified to one generator.
• Evaluate how understanding ideals generated by sets can aid in solving problems related to quotient rings and maximal ideals.
  • Understanding ideals generated by sets provides foundational knowledge needed to tackle more advanced topics like quotient rings and maximal ideals. By recognizing how to form ideals from sets, one can easily identify which elements belong to specific cosets in quotient rings. Additionally, this comprehension allows for discerning maximal ideals, as knowing which sets generate these critical structures helps solve problems related to ring homomorphisms and factorization within algebraic systems.

Related terms

Ring: A mathematical structure consisting of a set equipped with two binary operations, typically called addition and multiplication, satisfying certain properties.

Quotient Ring: A construction that forms a new ring from an existing ring by partitioning it into cosets of an ideal.

Maximal Ideal: An ideal that is proper (not equal to the whole ring) and is not contained in any other proper ideal.