How Do You Spell DIVISION RING?

Pronunciation: [dɪvˈɪʒən ɹˈɪŋ] (IPA)

A division ring is a mathematical term referring to a non-commutative ring in which every nonzero element has a multiplicative inverse. The spelling of the word is "dɪˈvɪʒən rɪŋ", with the emphasis on the first syllable. The word "division" is spelled with an "i" instead of an "a" because it comes from the Latin word "divisio," which means "division or separation." Meanwhile, "ring" is written as in the English language, but it means a specific type of algebraic structure, not a piece of jewelry.

DIVISION RING Meaning and Definition

1. A division ring, also known as a skew field, is a mathematical structure that extends the concept of a field. A division ring is a set equipped with two binary operations, addition and multiplication, which satisfy a set of axioms.

   In a division ring, the operation of addition turns it into an abelian group, meaning that it is commutative and possesses an identity element, as well as the existence of inverses for every element. The operation of multiplication in a division ring is also associative and distributive over addition.

   What distinguishes a division ring from a field is the existence of multiplicative inverses for all non-zero elements. In other words, every non-zero element of a division ring has a unique multiplicative inverse. This allows for a wider range of possible algebraic operations and inverses on the elements of the division ring.

   Unlike fields, division rings do not necessarily require the commutativity of multiplication. Therefore, a division ring may have non-commutative multiplication, making it more flexible and diverse in its structure. Division rings are often associated with non-commutative algebraic structures, providing a framework for various mathematical concepts and applications. They serve as essential tools in algebraic geometry, functional analysis, and representation theory, among other fields.

Etymology of DIVISION RING

The term "division ring" is derived from the mathematical operations and properties associated with the structure.

The word "ring" in mathematics refers to a set equipped with two binary operations, usually addition and multiplication, that satisfy specific axioms. In the case of a ring, the multiplication operation does not necessarily possess an inverse for all nonzero elements.

The term "division" originates from the property that a division ring possesses. In a division ring, every nonzero element has a multiplicative inverse, meaning that for any element "a" in the division ring, there exists another element "b" such that "ab" is equal to the multiplicative identity of the ring (usually denoted by 1).

So, combining the terms "division" and "ring", the name "division ring" is used to describe a mathematical structure where the multiplication operation has the property of division, allowing for the existence of multiplicative inverses for nonzero elements.