Division Ring

Definition

A division ring is a type of algebraic structure that consists of a set equipped with two binary operations, addition and multiplication, where every non-zero element has a multiplicative inverse. Unlike fields, division rings do not require multiplication to be commutative. This means that while division rings share many properties with fields, they allow for more flexibility in the arrangement of elements, particularly concerning multiplication.

5 Must Know Facts For Your Next Test

1. Every division ring is a ring, but not every ring is a division ring, as rings do not require the existence of multiplicative inverses for all non-zero elements.
2. The concept of a division ring includes examples such as the quaternions, which are non-commutative but still possess the necessary properties of a division ring.
3. In a division ring, the cancellation laws hold true; if $a eq 0$, then $ab = ac$ implies $b = c$.
4. Division rings can be characterized by their dimension over their center; if the center is a field, then the structure can be viewed as a vector space.
5. The existence of a non-zero element without an inverse within a division ring would violate its definition, ensuring that all non-zero elements must be invertible.

Review Questions

• What distinguishes a division ring from a field and why is this distinction significant?
  • The main distinction between a division ring and a field lies in the commutativity of multiplication. In a division ring, multiplication may not be commutative, while in a field it must be. This distinction is significant because it allows for greater variety in algebraic structures. For instance, quaternions serve as an example of a division ring that cannot form a field due to their non-commutative multiplication.
• How does the property of having multiplicative inverses for all non-zero elements shape the behavior of structures like division rings?
  • The property of having multiplicative inverses for all non-zero elements ensures that division rings maintain certain algebraic behaviors similar to those found in fields. This means that division rings allow for solutions to equations involving multiplication and enable operations such as dividing by non-zero elements. However, because the multiplication does not have to be commutative, this opens up possibilities for more complex structures and interactions among elements.
• Analyze the implications of the existence of division rings in broader mathematical contexts, particularly in relation to linear algebra and representation theory.
  • The existence of division rings has profound implications in areas such as linear algebra and representation theory because they serve as scalars for vector spaces. When considering vector spaces over division rings, one can explore representations that extend beyond traditional fields, allowing for diverse geometrical interpretations. This complexity enriches the study of linear transformations and leads to deeper understandings in both algebra and geometry, especially when dealing with non-commutative structures like quaternions or other specialized systems.

Related terms

Field: A field is a set with two operations, addition and multiplication, where every non-zero element has an inverse and multiplication is commutative.

Ring: A ring is an algebraic structure consisting of a set equipped with two binary operations satisfying properties like associativity and distributivity.

Multiplicative Inverse: The multiplicative inverse of an element is another element which, when multiplied together, yields the identity element for multiplication.