GP-INJECTIVE RINGS NEED NOT BE P-INJECTIVE.

A ring R is called left P-injective if for every a ∈ R , aR = r(l( a )) where l( ⋅ ) and r( ⋅ ) denote left and right annihilators respectively. The ring R is called left GP -injective if for any 0 ≠ a ∈ R , there exists n > 0 such that a n ≠ 0 and a n R = r(l( a n )). As a response to an open question on GP -injective rings, an example of a left GP -injective ring which is not left P -injective is given. It is also proved here that a ring R is left FP -injective if and only if every matrix ring 𝕄 n ( R ) is left GP -injective. [ABSTRACT FROM AUTHOR]