Let R be an associated ring with identity. A new equivalent characterization of pure projective left R-modules is given by applying homological methods. It is proved that a left R-module P is pure projective if and on...Let R be an associated ring with identity. A new equivalent characterization of pure projective left R-modules is given by applying homological methods. It is proved that a left R-module P is pure projective if and only if for any pure epimorphism E→M→0, where E is pure injective, HomR（P, E）→HomR（P, M）→0 is exact. Also, we obtain a dual result of pure injective left R-modules. Furthermore, it is shown that every pure projective left R-module is closed under pure submodule if and only if every pure injective left R-module is closed under pure epimorphic image.展开更多
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 d...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.展开更多
文摘Let R be an associated ring with identity. A new equivalent characterization of pure projective left R-modules is given by applying homological methods. It is proved that a left R-module P is pure projective if and only if for any pure epimorphism E→M→0, where E is pure injective, HomR（P, E）→HomR（P, M）→0 is exact. Also, we obtain a dual result of pure injective left R-modules. Furthermore, it is shown that every pure projective left R-module is closed under pure submodule if and only if every pure injective left R-module is closed under pure epimorphic image.
文摘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.
