2022/07/13 by Gheibi, Mohsen · 2 citations
#13D05 #13D07 #13H10 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2207.06170
Following our previous work about quasi-projective dimension, in this paper, we introduce quasi-injective dimension as a generalization of injective dimension. We recover several well-known results about injective and Gorenstein-injective dimensions in the context of quasi-injective dimension such as the following. (a) If the quasi-injective dimension of a finitely generated module M over a local ring R is finite, then it is equal to the depth of R. (b) If there exists a finitely generated module of finite quasi-injective dimension and maximal Krull dimension, then R is Cohen-Macaulay. (c) If there exists a nonzero finitely generated module with finite projective dimension and finite quasi-injective dimension, then R is Gorenstein. (d) Over a Gorenstein local ring, the quasi-injective dimension of a finitely generated module is finite if and only if its quasi-projective dimension is finite.