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.