Equivalent Definitions
A ring R is prime if and only if the zero ideal {0} is a prime ideal in the noncommutative sense.
This being the case, the equivalent conditions for prime ideals yield the following equivalent conditions for R to be a prime ring:
- For any two ideals A and B of R, AB={0} implies A={0} or B={0}.
- For any two right ideals A and B of R, AB={0} implies A={0} or B={0}.
- For any two left ideals A and B of R, AB={0} implies A={0} or B={0}.
Using these conditions it can be checked that the following are equivalent to R being a prime ring:
- All right ideals are faithful modules as right R modules.
- All left ideals are faithful left R modules.
Read more about this topic: Prime Ring
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