Fund Project:Supported by the Doctoral Fund of the Ministry of Education of China (Grant No.200803570003), the College Excellent Young Talents Fund of Anhui Province (Grant No.2013SQRL071ZD).
The definition of $AP$-injectivity motivates us to generalize the injectivity to almost wnil-injectivity and almost nil $n$-injectivity. The aim of this paper is to investigate characterizations and properties of almost wnil-injective rings and almost nil $n$-injective rings. Various results are developed, and many conclusions extend known results.