A simple domain not being skew field?

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SUMMARY

The discussion centers on the concept of simple domains and their relationship to fields and rings. A simple domain is defined as a domain that is also a simple ring, which is a ring without zero divisors. The example of a simple ring, M_n(F), where F is a field, is mentioned; however, it is clarified that M_n(F) does not qualify as a domain. The conversation emphasizes the challenge of identifying a simple integral domain that is not a field.

PREREQUISITES
  • Understanding of ring theory and definitions of rings and fields
  • Familiarity with the concept of zero divisors in algebra
  • Knowledge of simple rings and their properties
  • Basic comprehension of integral domains and their characteristics
NEXT STEPS
  • Research the properties of simple rings and their classifications
  • Explore examples of integral domains that are not fields
  • Study the implications of commutativity in ring theory
  • Learn about noncommutative rings and their applications
USEFUL FOR

Mathematicians, algebra students, and anyone interested in advanced ring theory and the distinctions between domains and fields.

peteryellow
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Can you find an example of a simple domain not being skew field?
 
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Like in your other thread: What are your thoughts on this? Can you list examples of simple rings? Domains?

What if we assume commutativity, i.e. can you find a simple integral domain that isn't a field? You shouldn't - but this might shed some light on the noncommutative case.
 
Simple ring M_n(F) where F is a field, but what is the definition of a simple domain?
 
A domain is a ring without zero divisors, i.e. xy=0 implies either x=0 or y=0. A simple domain is a domain that is a simple ring.

Unfortunately M_n(F) isn't a domain, so you're going to have to be more creative if you want to come up with an example!
 

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