A simple domain not being skew field?

[peteryellow]

Can you find an example of a simple domain not being skew field?

 

[morphism]

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.

 

[peteryellow]

Simple ring M_n(F) where F is a field, but what is the definition of a simple domain?

 

[morphism]

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!