The Definition And Proportion Of I-injective Modules

Let A be a commutative Noetherian ring, E an A-module and I an ideal of A such that ht(I) > 0. E is an I-injective module, if Supp (Ext_A~1 (M, E)) (?) V(I),for any finitely generated .A-module M. The proposition that Supp (Ext_A~1(A/p, E)) (?) V(I), for any p ∈ Spec(A) if and only if E is an I-injective module has also been proved in this article.And we claimed that the localization of E at (?)p (?) V(I) is injective.It is clear that any injective module is I-injective, but the contrary is not true. An example for the existence of an I-injective module is given. The definition of I-injective dimension, I-projective module, and I-projective dimension are introduced and characterized. Moreover, the global I-dimension of the ring is also defined.In this article, the proportion of I-projective dimension has been payed more attention, which has been characterized by means of the homological functor, 'Tor'. It has been proved that if (A, m) is a commutative Noetherian local ring, and the global I-dimension of A is finite, then for any finitely generated A-module M, there exists a prime ideal p (?) V(I) such that pd_I(M) + f-depth_I(p, M) ≥D_I(A)...