ON SMALL INJECTIVE RINGS AND MODULES.

A right R-module M_R is called small injective if every homomorphism from a small right ideal to M_R can be extended to an R-homomorphism from R_R to M_R. A ring R is called right small injective, if the right R-module R_R is small injective. We prove that R is semiprimitive if and only if every simple right (or left) R-module is small injective. Further we show that the Jacobson radical J of a ring R is a noetherian right R-module if and only if, for every small injective module E_R, E^(ℕ) is small injective. [ABSTRACT FROM AUTHOR]