Rings

Definition

In abstract algebra, a ring is a set equipped with two binary operations, typically called addition and multiplication, that generalizes the arithmetic of integers. These operations must satisfy certain properties, such as associativity, distributivity, and the presence of an additive identity. Rings are essential structures in mathematics as they provide a framework for studying algebraic systems and their homomorphisms and isomorphisms.

5 Must Know Facts For Your Next Test

1. A ring must have two operations: addition and multiplication, where addition must form an abelian group and multiplication must be associative.
2. Rings can be classified into different types, such as commutative rings (where multiplication is commutative) and non-commutative rings.
3. The presence of an identity element for multiplication distinguishes unital rings from general rings.
4. Homomorphisms between rings preserve the structure of the rings and play a crucial role in understanding their relationships.
5. The study of ideals in rings leads to the creation of quotient rings, which helps to simplify complex ring structures.

Review Questions

• How do the properties of addition and multiplication in a ring relate to the concept of homomorphisms?
  • In a ring, addition must form an abelian group while multiplication is associative, allowing for the structure to be preserved under homomorphisms. A homomorphism maps elements from one ring to another while maintaining these operations' properties. Therefore, when a homomorphism exists between two rings, it ensures that the sum and product of mapped elements correspond correctly in both rings, reinforcing the connections between their algebraic structures.
• Compare and contrast rings and fields, focusing on their definitions and key differences.
  • Rings are sets with two operations (addition and multiplication) that adhere to specific properties but do not require every non-zero element to have a multiplicative inverse. In contrast, fields are a special type of ring where every non-zero element indeed possesses an inverse for multiplication, allowing division by non-zero elements. This distinction makes fields more structured than rings, allowing them to support more advanced algebraic manipulations.
• Evaluate the significance of ideals in the study of rings and their homomorphisms.
  • Ideals are pivotal in understanding rings because they allow for the construction of quotient rings, which simplify complex ring structures into more manageable forms. By analyzing how ideals interact with homomorphisms, we can gain insight into the relationships between different rings. Ideals help us explore concepts like factorization and divisibility within rings, ultimately leading to deeper results such as the structure theorem for finitely generated modules over a principal ideal domain.

Related terms

Field:

A field is a ring in which every non-zero element has a multiplicative inverse, allowing for division except by zero.

Ideal:

An ideal is a special subset of a ring that absorbs multiplication by elements from the ring and is used to construct quotient rings.

Homomorphism:

A homomorphism is a structure-preserving map between two algebraic structures, such as rings, that respects the operations defined on those structures.