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
Famous quotes containing the words equivalent and/or definitions:
“To place oneself in the position of God is painful: being God is equivalent to being tortured. For being God means that one is in harmony with all that is, including the worst. The existence of the worst evils is unimaginable unless God willed them.”
—Georges Bataille (1897–1962)
“What I do not like about our definitions of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”
—G.C. (Georg Christoph)