On a Generalization of Injective Rings

Abstract

A ring R is called left IP-injective if every homomorphism from a left ideal of R into R with principal image is given by right multiplication by an element of R. It is shown that R is left IP-injective if and only if R is left P-injective and left GIN (i.e., r(I ∩ K) = r(I) + r(K) for each pair of left ideals I and K of R with I principal). We prove that R is QF if and only if R is right noetherian and left IP-injective if and only if R is left perfect, left GIN and right simple-injective. We also show that, for a right CF left GIN-ring R, R is QF if and only if Soc(R _R ) ⊆ Soc( _R R). Two examples are given to show that an IP-injective ring need not be self-injective and a right IP-injective ring is not necessarily left IP-injective respectively.

Key Words:

• IP-injective ring
• GIN ring
• QF ring

2000 Mathematics Subject Classification:

• 16D50
• 16P40

Acknowledgments

Part of this work was carried out during a visit by the second author to the Ohio State University at Lima. He is grateful to the members of the Mathematics Department for their kind hospitality. This research was supported in part by NNSF of China (No. 10171011 and 10071035), NSF of Jiangsu Province (No. BK 2001001) and by the Ohio State University.