Some relations about the generalizations of self-injective ring: P-injective ring, GP-injective ring, AP-injective ring, simple-injective ring and n-injective ring are studied.

We prove in this paper that R and an finite simple extension of R have self-injectivity simultaneously, and based on this,give a new proof for the result that every weakly invertible linear finite automaton over R with delay τ has a linear weak inverse with delay τ if and only if R is a self-injective ring.