REMARKS ON PRIME IDEALS AND REDUCED RINGS

We first prove that a ring is prime if and only if whenever for a nonzero proper ideal of and . Then, by using this result, we give another proof of the well known fact that is prime if and only if the polynomial ring over is prime, We also prove that if is an ideal of maximal with respect to the property that is disjoint from , then is a prime ideal of , where is a ring and is an -system for a nonzero proper ideal of . A ring is called trivial if has no proper subrings with the identity of . It is proved that a trivial ring is isomorphic to either (when ), (when and the primary decomposition of is square free), or (when and the primary decomposition of is not square free), where is the characteristic of .