Study of hopficity in certain classes of rings

The main results proved in this paper are: 1. For any non-zero vector space VD over a division ring D, the ring R = En d(V-D) is hopfian as a ring. 2. Let R be a reduced pi-regular ring and B(R) the boolean ring of idempote nts of R. IIB(R) is hopfian so is R. The converse is not true even when R i s strongly regular. 3. Let X be a completely regular space, C(X) (resp. CI(X)) the ring of real valued (resp. bounded real valued) continuous functions on X. Let R be any one of C(X) or C*(X). Then R is an exchange ring if and only if X is zero dimensional in the sense of Katetov. For any infinite compact totally disco nnected space X, C(X') is an exchange ring which is not von Neumann regular . 4. Let R be a reduced commutative exchange ring. If R is hopfian so is the polynomial ring R[T-1,...,T-n] in n commuting indeterminates over R, where n is any integer greater than or equal to 1. 5. Let n be a reduced exchange ring. If R is hopfian so is the polynomial r ing R[T].