摘要
Let C be a semidualizing module for a commutative ring R. It is shown that the ：IC-injective dimension has the ability to detect the regularity of R as well as the Pc-projective dimension. It is proved that if D is dualizing for a Noetherian ring R such that idR（D） = n 〈 ∞, then ：ID-idR（F） ≤ n for every flat R-module F. This extends the result due to Enochs and Jenda. Finally, over a Noetherian ring R, it is shown that if M is a pure submodule of an R-module N, then/TC-idR（M） ≤ IC-idR（N）. This generalizes the result of Enochs and Holm.
Let C be a semidualizing module for a commutative ring R. It is shown that the ：IC-injective dimension has the ability to detect the regularity of R as well as the Pc-projective dimension. It is proved that if D is dualizing for a Noetherian ring R such that idR（D） = n 〈 ∞, then ：ID-idR（F） ≤ n for every flat R-module F. This extends the result due to Enochs and Jenda. Finally, over a Noetherian ring R, it is shown that if M is a pure submodule of an R-module N, then/TC-idR（M） ≤ IC-idR（N）. This generalizes the result of Enochs and Holm.
