Rings in which every ideal is pure-projective or FP-projective.

A ring R is said to be left pure-hereditary (resp. RD -hereditary) if every left ideal of R is pure-projective (resp. RD -projective). In this paper, some properties and examples of these rings, which are nontrivial generalizations of hereditary rings, are given. For instance, we show that if R is a left RD -hereditary left nonsingular ring, then R is left Noetherian if and only if u . dim ( R R ) < ∞ . Also, we show that a ring R is quasi-Frobenius if and only if R is a left FGF, left coherent right pure-injective ring. A ring R is said to be left FP-hereditary if every left ideal of R is FP-projective. It is shown that if R is a left CF ring, then R is left Noetherian if and only if R is left pure-hereditary, if and only if R is left FP-hereditary, if and only if R is left coherent. It is shown that every left self-injective left FP-hereditary ring is semiperfect. Finally, it is shown that a ring R is left FP-hereditary (resp. left coherent) if and only if every submodule (resp. finitely generated submodule) of a projective left R -module is FP-projective, if and only if every pure factor module of an injective left R -module is injective (resp. FP-injective), if and only if for each FP-injective left R -module U , E ( U ) / U is injective (resp. FP-injective). [ABSTRACT FROM AUTHOR]