division ring: meaning, definition, pronunciation and examples
Very Low (Technical)Formal / Academic (Mathematical)
Quick answer
What does “division ring” mean?
A ring in which every nonzero element has a multiplicative inverse.
Audio
Pronunciation
Definition
Meaning and Definition
A ring in which every nonzero element has a multiplicative inverse.
In algebra, a division ring is a set with addition, subtraction, multiplication, and division (except by zero) operations where multiplication is not necessarily commutative. A commutative division ring is a field.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage between UK and US mathematical communities. Both use the standard definition.
Connotations
Purely technical, with no regional connotations.
Frequency
Identically low frequency in both academic contexts.
Grammar
How to Use “division ring” in a Sentence
[The/This/An] division ring [is/has/provides]division ring [over/of] [a set/field]Vocabulary
Collocations
Examples
Examples of “division ring” in a Sentence
adjective
British English
- The division-ring structure is fundamental to the proof.
American English
- The division ring axioms were satisfied.
Usage
Meaning in Context
Business
Never used in business contexts.
Academic
Exclusive to advanced undergraduate and graduate mathematics, particularly abstract algebra and ring theory.
Everyday
Virtually unknown in everyday conversation.
Technical
Core term in the technical lexicon of pure mathematics.
Vocabulary
Synonyms of “division ring”
Vocabulary
Antonyms of “division ring”
Watch out
Common Mistakes When Using “division ring”
- Pronouncing 'ring' as the jewellery (incorrect stress). The correct pronunciation treats both words equally: 'division RING'.
- Using it outside of its strict mathematical definition.
- Confusing it with a 'division algebra', which requires a base field.
FAQ
Frequently Asked Questions
Yes. A field is a commutative division ring, so it is a specific type of division ring.
The quaternions, discovered by Hamilton, are a classic example. Here, i × j = k, but j × i = -k, so multiplication is not commutative.
A ring only requires addition and multiplication. A division ring is a special kind of ring where you can also divide by any nonzero element (i.e., every nonzero element has a multiplicative inverse).
The term 'skew field' emphasises that it is like a field ('body' in some languages) but with multiplication 'skewed' or twisted away from commutativity.
Division ring is usually formal / academic (mathematical) in register.
Division ring: in British English it is pronounced /dɪˈvɪʒ.ən rɪŋ/, and in American English it is pronounced /dɪˈvɪʒ.ən rɪŋ/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of a 'ring' where you can 'divide' (except by zero), but the multiplication might be 'skewed' (non-commutative).
Conceptual Metaphor
A GENERALISED ARITHMETIC SYSTEM: A division ring is metaphorically a playground for numbers where all the usual rules of arithmetic apply, except that a × b might not equal b × a.
Practice
Quiz
Which of the following is a key property of a division ring?