Abstract
In this note we study rings with ascending and descending chain conditions on small right ideals. We show that these classes are closed under finite direct sum and quotients for every non-zero small ideal, but they are not closed under extensions or taking ideals, however if I is a small ideal in a commutative ring R and if I and R/I satisfies the a. c. c. (resp. d.c.d.) on small ideals, then R satisfies the a. c. c. (resp. d. c. c.) on small ideals. Finally we show that Camillo's theorem need not be true in non-commutative rings. Then we prove that a ring is Noetherian if and only if every quotient satisfies the a. c. c. on complement and small ideals.