| Summary: | A ring R is said to be prime if AB = 0 implies A= 0 or B = 0 for any (two sided) ideals A, B of R. If R is commutative, this condition is equivalent to: ab= 0 implies a= 0 or b = 0, where a and b are arbitrary elements of R. Thus it is seen that a prime ring is a generalization of an integral domain, and hence merits investigation. Prime rings with certain chain conditions were studied by Goldie [4] and Johnson [6]. Using different methods, we shall establish their main results. (In Goldie’s case a result similar to his structure theorem will be proved. ) If A, B, C are right R-modules such that A<B, we write A< B(C) if every R-homomorphism from A into C can be extended uniquely to an irreducible partial R-homomorphism from B into C [3].
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