A simple domain not being skew field?

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Discussion Overview

The discussion revolves around the concept of simple domains and their relationship to skew fields. Participants explore examples and definitions related to simple rings and domains, particularly focusing on whether a simple domain can exist that is not a skew field.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant asks for an example of a simple domain that is not a skew field.
  • Another participant suggests considering commutativity and challenges the notion of finding a simple integral domain that isn't a field, implying that such an example may not exist.
  • A participant mentions the simple ring M_n(F) where F is a field, questioning the definition of a simple domain.
  • It is clarified that a domain is defined as a ring without zero divisors, and a simple domain is a domain that is also a simple ring. However, M_n(F) is noted not to be a domain, indicating a need for more creative examples.

Areas of Agreement / Disagreement

Participants express differing views on the existence of simple domains that are not skew fields, with some suggesting that such examples may not exist while others seek clarification and examples.

Contextual Notes

There is an unresolved discussion regarding the definitions and properties of simple domains and rings, particularly in relation to commutativity and the characteristics of specific examples like M_n(F).

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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