Editing Prime ring (section)

==Equivalent definitions==
A ring ''R'' is prime [[if and only if]] the [[zero ideal]] {0} is a [[Prime_ideal#Prime_ideals_for_noncommutative_rings|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 [[ideal (ring theory)|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 nonzero right ideals are [[faithful module|faithful]] as right ''R''-modules.
*All nonzero left ideals are faithful as left ''R''-modules.