Let R be a ring, a right ideal I of R is called small if for every proper right ideal K of R, I +K = R. A ring R is called right small injective if every homomorphism from a small right ideal to R R can be extended to an R-homomorphism from R R to R R. Properties of small injective rings are explored and several new characterizations are given for QF rings and P F rings, respectively.

A right ideal A of a ring R is called annihilator-small if A+ T = R; T a right ideal, implies that l(T ) = 0; where l( ) indicates the left annihilator. The sum Ar of all such right ideals turns out to be a two-sided ideal that contains the Jacobson radical and the left singular ideal, and is contained in the ideal generated by the total of the ring. The ideal Ar is studied, conditions when it ...

let r be a right gf-closed ring with finite left and right gorenstein global dimension. we prove that if i is an ideal of r such that r/i is a semi-simple ring, then the gorensntein flat dimensnion of r/i as a right r-module and the gorensntein injective dimensnnion of r/i as a left r-module are identical. in particular, we show that for a simple module s over a commutative gorensntein ring r, ...